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arxiv: 1907.05429 · v1 · pith:PCY6DOTDnew · submitted 2019-07-11 · ✦ hep-th

Spherical Contours, IR Divergences and the geometry of Feynman parameter integrands at one loop

Akshay Yelleshpur Srikant This is my paper

Pith reviewed 2026-05-24 22:48 UTC · model grok-4.3

classification ✦ hep-th
keywords spherical contoursFeynman parametersIR divergencesleading singularitiesone-loop integralsN=4 SYMloop integrands
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The pith

Spherical contours translate the discontinuity across branch cuts into Feynman parameter space to define leading singularities and extract one-loop IR divergences directly there.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that spherical contours capture branch-cut discontinuities when transplanted to Feynman parameter space. This construction produces a parameter-space version of leading singularities, which in turn supplies a procedure for writing down the Feynman parameter integrand itself without ever consulting the original momentum-space loop integrand. The same contours also isolate the leading infrared divergences of one-loop graphs straight from the parameter integrand. The method is applied to several examples, including graphs in N=4 super Yang-Mills, to illustrate geometric features of the resulting integrands. A reader would care because the approach offers an alternative route to loop-integral data that stays inside the Feynman-parameter representation throughout.

Core claim

Spherical contours introduced in prior work translate the concept of discontinuity across a branch cut to Feynman parameter space. These 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 with no reference to the momentum space loop integrand. The same contours connect directly to the computation of leading IR divergences of one-loop graphs in Feynman parameter space, and the geometry of the resulting integrands is explored in N=4 SYM.

What carries the argument

Spherical contours that capture the discontinuity across a branch cut when placed in Feynman parameter space.

If this is right

  • Leading IR divergences of one-loop graphs become extractable directly from the Feynman-parameter integrand via the spherical contours.
  • The Feynman-parameter integrand for a given graph can be written down without first constructing or consulting the momentum-space loop integrand.
  • An analog of leading singularities exists entirely inside Feynman parameter space.
  • Geometric properties of the integrands in N=4 SYM can be read off from the spherical-contour residues.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the contour prescription continues to work at two loops, the same construction could supply leading singularities for higher-loop parameter integrands.
  • The method may intersect with existing parameter-space techniques such as sector decomposition or differential equations by providing an independent geometric check on residues.
  • The observed geometry in N=4 SYM might indicate which rational functions of parameters survive in other theories with similar infrared structure.

Load-bearing premise

Spherical contours correctly reproduce the discontinuity across a branch cut once they are moved into Feynman parameter space.

What would settle it

A direct calculation of the leading IR pole or residue of a known one-loop graph using only the spherical-contour prescription in parameter space that fails to match the value obtained from the standard momentum-space result would falsify the method.

read the original abstract

Spherical contours introduced in \cite{SphericalContours} 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 with no reference to the momentum space loop integrand. Finally, we explore some interesting features of Feynman parameter integrands in $\mathcal{N}=4$ SYM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that spherical contours introduced in prior work can be translated into Feynman parameter space to capture discontinuities across branch cuts. This enables direct computation of leading IR divergences for one-loop graphs in parameter space, the construction of a Feynman parameter analog of leading singularities, and a method to determine the Feynman parameter integrands without reference to the momentum-space loop integrand. The paper also examines geometric features of these integrands in N=4 SYM.

Significance. If the contour translation preserves the correct discontinuities and residues, the work would supply a geometric, parameter-space framework for extracting IR divergences and constructing integrands independently of momentum space. This could streamline analyses in gauge theories and provide new tools for one-loop calculations in N=4 SYM, building explicitly on the cited contour construction.

major comments (2)
  1. [Section on contour translation (following the abstract's reference to SphericalContours)] The section defining the translation of spherical contours to Feynman parameter space: the central claim that these contours correctly capture branch-cut discontinuities (and thereby enable IR divergence extraction) rests on the imported definition from the cited reference without an explicit check that the transplanted contours enclose the correct poles or reproduce known IR residues for standard one-loop graphs such as the bubble integral. This verification is load-bearing for the subsequent construction of the leading-singularity analog and the 'no reference to momentum space' method.
  2. [Section on leading-singularity analog and integrand determination] The section presenting the method for determining Feynman parameter integrands: the assertion that integrands can be obtained with no reference to the momentum-space loop integrand is undermined if the locations of singularities in the parameter simplex are determined using information imported from momentum space; an explicit demonstration that the spherical-contour geometry alone suffices is required.
minor comments (2)
  1. [Abstract] Abstract: the phrasing 'develop a method of determining Feynman parameter integrands with no reference to the momentum space loop integrand' is repeated in the body; a single concise statement would improve readability.
  2. [Throughout] Notation for contours and parameters should be introduced once with a clear table or list to avoid redefinition across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Section on contour translation (following the abstract's reference to SphericalContours)] The section defining the translation of spherical contours to Feynman parameter space: the central claim that these contours correctly capture branch-cut discontinuities (and thereby enable IR divergence extraction) rests on the imported definition from the cited reference without an explicit check that the transplanted contours enclose the correct poles or reproduce known IR residues for standard one-loop graphs such as the bubble integral. This verification is load-bearing for the subsequent construction of the leading-singularity analog and the 'no reference to momentum space' method.

    Authors: We agree that an explicit verification strengthens the foundation. The translation follows directly from the definition in the cited work, but we will add a dedicated subsection verifying the bubble integral. This will explicitly show that the spherical contours in Feynman parameter space enclose the correct poles and reproduce the known leading IR residue, confirming the discontinuities are captured correctly before proceeding to the leading-singularity analog. revision: yes

  2. Referee: [Section on leading-singularity analog and integrand determination] The section presenting the method for determining Feynman parameter integrands: the assertion that integrands can be obtained with no reference to the momentum-space loop integrand is undermined if the locations of singularities in the parameter simplex are determined using information imported from momentum space; an explicit demonstration that the spherical-contour geometry alone suffices is required.

    Authors: The construction determines singularity locations solely from the intersections of the spherical contours with the boundaries of the Feynman parameter simplex; momentum-space data enters only as illustrative examples. To make this fully explicit, we will add a self-contained example in which the integrand is recovered using only the contour geometry and simplex combinatorics, with no momentum-space input used to locate the singularities. revision: yes

Circularity Check

1 steps flagged

Central construction depends on spherical contour definition imported from prior self-citation without re-derivation

specific steps
  1. self citation load bearing [Abstract]
    "Spherical contours introduced in [SphericalContours] 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 with no reference to the momentum space loop integrand."

    The translation property that enables the entire construction (discontinuity capture in parameter space, leading to integrand determination without momentum-space input) is taken as given from the cited prior work. Subsequent claims of an analog of leading singularities and direct IR divergence computation therefore inherit their validity from that external definition rather than deriving or validating the contour-residue correspondence independently in this paper.

full rationale

The paper's method for determining Feynman parameter integrands and extracting IR divergences rests on the translation of branch-cut discontinuities via spherical contours, which is introduced and justified solely by citation to prior work. This matches self-citation load-bearing because the load-bearing premise (that the contours correctly capture the required residues in parameter space) reduces to the cited definition rather than an independent derivation or external validation within this manuscript. No fitted-input or self-definitional reductions appear in the provided abstract or claims. The result is moderate circularity burden (score 4) rather than full collapse, as the paper still performs new connections to one-loop IR poles.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the text provided.

pith-pipeline@v0.9.0 · 5637 in / 1274 out tokens · 20858 ms · 2026-05-24T22:48:00.498274+00:00 · methodology

discussion (0)

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Reference graph

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