Spherical contours, IR divergences and the geometry of Feynman parameter integrands at one loop
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Springer
Received: September 2, Revised: May 24, Accepted: June 10, Published: July 31,
2019 2020 2020 2020
Akshay Yelleshpur Srikant Department of Physics, Princeton University, Washington Road, Princeton, NJ 08540, U.S.A.
E-mail: [email protected] 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 N = 4 SYM. Keywords: 1/N Expansion, Scattering Amplitudes ArXiv ePrint: 1907.05429
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)236
JHEP07(2020)236
Spherical contours, IR divergences and the geometry of Feynman parameter integrands at one loop
Contents 1 Introduction
1
2 Feynman parametrization revisited
2 4 7 8 9 11
4 Algebraic aspects of spherical residues 4.1 Properties of Feynman integrals coming from loop integrals 4.2 Spherical contours meet IR divergences
12 14 15
5 Constructing integrands using spherical residues 5.1 5 point integrands 5.2 6 point integrands
16 16 17
6 Feynman parametrization in planar N = 4 SYM
18
7 Outlook
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A Cuts of Feynman integrals
20
B Spherical contour with a quadratic numerator
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C Leading singularities at 6 points
22
D Feynman parametrizing the MHV planar 1-loop integrand
23
1
Introduction
A connection between the singularity structure of one-loop integrals and the projective geometry of their associated Feynman parameter integrand was established in [1]. One of the central results of this paper was the introduction of a new kind of residue in Feynman parameter space, associated with “spherical contours” which were operations involving only the one-loop integrand. These capture information about discontinuities of the integrals across various branch cuts. It was shown that this calculus based operation also has an algebraic interpretation. The purpose of this paper is to provide some additional details
–1–
JHEP07(2020)236
3 1-loop IR divergences 3.1 Composite residues in momentum space 3.2 Composite residues in Feynman parameter space 3.3 Proof for general one-loop integrals 3.4 A basis for IR finite integrals in Feynman parameter space
2
Feynman parametrization revisited
Although Feynman parametrization is a familiar trick, let us begin by discussing it in a more geometric way. This will highlight some of the features of Feynman parameter integrals which are important for the rest of the paper. Consider the scalar, one-loop integrals of the form (µ2 is the mass scale introduced in dimensional regula
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