Abstract
Spherical contours introduced in [1] translate the concept of “discontinuity across a branch cut” to Feynman parameter space. In this paper, we further explore spherical contours and connect them to the computation of leading IR divergences of 1 loop graphs directly in Feynman parameter space. These spherical contours can be used to develop a Feynman parameter space analog of “Leading Singularities” of loop integrands which allows us to develop a method of determining Feynman parameter integrands without the need to sum over Feynman diagrams in momentum space. Finally, we explore some interesting features of Feynman parameter integrands in mathcal{N} = 4 SYM.
Highlights
On this algebraic interpretation and explore IR divergent integrals as the authors of [1] largely focused on finite integrals
We further explore spherical contours and connect them to the computation of leading IR divergences of 1 loop graphs directly in Feynman parameter space
These spherical contours can be used to develop a Feynman parameter space analog of “Leading Singularities” of loop integrands which allows us to develop a method of determining Feynman parameter integrands without the need to sum over Feynman diagrams in momentum space
Summary
Feynman parametrization is a familiar trick, let us begin by discussing it in a more geometric way. Where U and F are functions of the Feynman parameters xj and the external momenta and depend on the particular integral being evaluated They are connected to the propagators appearing in eq (2.1) via the polynomial n. The few paragraphs serve to provide intuition for analyzing IR divergences in Feynman parameter space It is well known [14] that IR divergences arise when the loop momentum becomes collinear with an external massless momentum pi, i.e. This should be equal to the coefficient of in dimensional regularization which is the cusp anomalous dimension Γ Note that this is purely a conjecture at this point and, we will demonstrate that this region in Feynman parameter space captures the essential information about the IR divergent region and can be used to calculate Γ
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