Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral

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 nontrivial Calabi-Yau manifold. We show that its differential equation can be cast into an ϵ-factorized form. This allows us to obtain the solution to any desired order in the dimensional regularization parameter ϵ. The method generalizes 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.