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.
Highlights
Ubiquity follows from integral reduction combined with the fact that any one-loop Feynman integral are believed to be expressible in terms of generalized polylogarithms
We explore the correspondence between one-loop Feynman integrals and simplicial geometry to describe the all-mass case: integrals with generic external and internal masses
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
Summary
In0 as defined in (1.2) is an n-dimensional integral with n loop-dependent factors in its denominator It has leading singularities: residues of maximal co-dimension. The integral In0 is positive definite (on the principal branch) for real kinematics, In may not be: for example, when det G0 < 0, our definition of In will be pure imaginary This is a convention; we could have chosen instead to use | det G0| in the normalization of (1.10), but the choice we have made is the more standard one (and the one we find will allow for slightly simpler formulas below). Even fixing branch conventions for det G, multidimensional residues are intrinsically oriented quantities whose signs depend on the orientation of the contour integral (or the ordering of integration variables in the Jacobian) that defines them. This conformal invariance can be better understood from the viewpoint of the embedding formalism, which we discuss in more detail in appendix A
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