Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral

TL;DR

This work addresses computing Feynman integrals tied to Calabi–Yau geometries by casting their differential equations into an $\varepsilon$-factorised form. Focusing on the four-loop equal-mass banana integral, it employs a mirror-map change of variables $(y\to \tau, q)$ and a redefinition of master integrals to incorporate a Calabi–Yau–specific function $K$, yielding an explicit $\varepsilon$-factorised system $dI = 2\pi i\,\varepsilon A_\tau I\, d\tau$ with a nine-letter alphabet. The authors provide boundary data at $y=0$ via Mellin–Barnes techniques and an analytic expression for $I_2$ up to $\mathcal{O}(\varepsilon^5)$, along with a $q$-expansion enabling fast numerical evaluation and cross-checks against \texttt{pySecDec}. This demonstrates that Calabi–Yau Feynman integrals can be solved to all orders in $\varepsilon$ and suggests a path to handling higher-dimensional Calabi–Yau geometries in loop calculations.

Abstract

Certain Feynman integrals are associated to Calabi-Yau geometries. We demonstrate how these integrals can be computed with the method of differential equations. The four-loop equal-mass banana integral is the simplest Feynman integral whose geometry is a non-trivial Calabi-Yau manifold. We show that its differential equation can be cast into an $\varepsilon$-factorised form. This allows us to obtain the solution to any desired order in the dimensional regularisation parameter $\varepsilon$. The method generalises to other Calabi-Yau Feynman integrals. Our calculation also shows that the four-loop banana integral is only minimally more complicated than the corresponding Feynman integrals at two or three loops.

Figures (3)

• Figure 1: The four-loop banana graph.
• Figure 2: Comparison of our result for $I_2^{(4)}$ (plotted as curves) with numerical results from pySecDec (points).
• Figure 3: Comparison of our result for $I_2^{(5)}$ (plotted as curves) with numerical results from pySecDec (points).