All-Mass $n$-gon Integrals in $n$ Dimensions

Abstract

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.

arXiv comment49 pages, 8 figures

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• Implications of the Landau Equations for Iterated Integrals
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• The Two-Loop Remainder Function for Eight and Nine Particles
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• Integrability for Feynman Integrals
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• Alphabet of one-loop Feynman integrals
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• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• Triangular tessellations of one-loop scattering amplitudes in ${\varphi }^3$ theory
• The Stratification of Rigidity