Bananas of equal mass: any loop, any order in the dimensional regularisation parameter

TL;DR

The paper presents a systematic, geometry-driven method to compute the $l$-loop equal-mass banana Feynman integrals to any order in the dimensional regularisation parameter $\varepsilon$ by transforming their differential equations into an $\varepsilon$-factorised form. Central to the approach is an ansatz for a master-integral basis informed by Calabi–Yau operator duality and self-duality, together with boundary data obtained from Mellin–Barnes analysis. The authors demonstrate the method explicitly for five- and six-loop cases, obtaining $\varepsilon$-expansions that include multiple zeta values and iterated integrals in terms of periods $I(\cdot)$, and they validate results against numerical benchmarks such as pySecDec. They also discuss constraints on the choice of periods and the appearance of nontrivial $Y$-invariants, showing that the CY structure saturates by four loops and that the symbol alphabet can be read off from the differential equation. Overall, the work provides a scalable, systematic framework for CY-related Feynman integrals and their $\varepsilon$-expansions across arbitrary loop orders.

Abstract

We describe a systematic approach to cast the differential equation for the $l$-loop equal mass banana integral into an $\varepsilon$-factorised form. With the known boundary value at a specific point we obtain systematically the term of order $j$ in the expansion in the dimensional regularisation parameter $\varepsilon$ for any loop $l$. The approach is based on properties of Calabi-Yau operators, and in particular on self-duality.

Figures (3)

• Figure 1: The graphs for the banana integrals from one to four loops.
• Figure 2: Comparison of our result for $M_1^{(5,5)}$ and $M_1^{(5,6)}$ at five loops with numerical results from pySecDec.
• Figure 3: Comparison of our result for $M_1^{(6,6)}$ and $M_1^{(6,7)}$ at six loops with numerical results from pySecDec.