All-mass n-gon integrals in n dimensions
Jacob L. Bourjaily, Einan Gardi, Andrew J. McLeod, Cristian Vergu
Abstract
A bstract We explore the correspondence between one-loop Feynman integrals and (hyperbolic) simplicial geometry to describe the all-mass case: integrals with generic external and internal masses. Specifically, we focus on n -particle integrals in exactly n space-time dimensions, as these integrals have particularly nice geometric properties and respect a dual conformal symmetry. In four dimensions, we leverage this geometric connection to give a concise dilogarithmic expression for the all-mass box in terms of the Murakami-Yano formula. In five dimensions, we use a generalized Gauss-Bonnet theorem to derive a similar dilogarithmic expression for the all-mass pentagon. We also use the Schläfli formula to write down the symbol of these integrals for all n . Finally, we discuss how the geometry behind these formulas depends on space-time signature, and we gather together many results related to these integrals from the mathematics and physics literature.