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One-Loop Integrals from Spherical Projections of Planes and Quadrics

Nima Arkani-Hamed, Ellis Ye Yuan

TL;DR

This work establishes a geometric framework for one-loop integrals by mapping singularities to spherical projections of quadrics and planes in projective Feynman-parameter space. It develops two complementary, internal algorithms—one based on differentiation and one on discontinuities via spherical contours—that jointly determine the symbol of any one-loop integral in arbitrary dimensions. The authors demonstrate a manifestly Lorentz-invariant connection between unitarity cuts and spherical-contour residues, and provide explicit symbol structures for broad families of scalar and tensor integrals, including degenerate quadrics and cases with linear factors in the denominator. The approach unifies Aomoto polylogarithms and standard polylogarithms under a common geometric language and offers a constructive route to compute or constrain the transcendental weight and symbol content of one-loop amplitudes, with clear implications for perturbative unitarity and analytic structure in quantum field theory.

Abstract

We initiate a systematic study of one-loop integrals by investigating the connection between their singularity structures and geometric configurations in the projective space associated to their Feynman parametrization. We analyze these integrals by two recursive methods, which leads to two independent algebraic algorithms that determine the symbols of any one-loop integrals in arbitrary spacetime dimensions. The discontinuities of Feynman diagrams are shown to arise from taking certain "spherical contour" residues in Feynman parameter space, which is geometrically interpreted as a projection of the quadric surface (associated to the Symanzik polynomial at one loop) through faces of the integration region (which is a simplex). This geometry also leads to a manifestly Lorentz-invariant understanding for perturbative unitarity at one loop.

One-Loop Integrals from Spherical Projections of Planes and Quadrics

TL;DR

This work establishes a geometric framework for one-loop integrals by mapping singularities to spherical projections of quadrics and planes in projective Feynman-parameter space. It develops two complementary, internal algorithms—one based on differentiation and one on discontinuities via spherical contours—that jointly determine the symbol of any one-loop integral in arbitrary dimensions. The authors demonstrate a manifestly Lorentz-invariant connection between unitarity cuts and spherical-contour residues, and provide explicit symbol structures for broad families of scalar and tensor integrals, including degenerate quadrics and cases with linear factors in the denominator. The approach unifies Aomoto polylogarithms and standard polylogarithms under a common geometric language and offers a constructive route to compute or constrain the transcendental weight and symbol content of one-loop amplitudes, with clear implications for perturbative unitarity and analytic structure in quantum field theory.

Abstract

We initiate a systematic study of one-loop integrals by investigating the connection between their singularity structures and geometric configurations in the projective space associated to their Feynman parametrization. We analyze these integrals by two recursive methods, which leads to two independent algebraic algorithms that determine the symbols of any one-loop integrals in arbitrary spacetime dimensions. The discontinuities of Feynman diagrams are shown to arise from taking certain "spherical contour" residues in Feynman parameter space, which is geometrically interpreted as a projection of the quadric surface (associated to the Symanzik polynomial at one loop) through faces of the integration region (which is a simplex). This geometry also leads to a manifestly Lorentz-invariant understanding for perturbative unitarity at one loop.

Paper Structure

This paper contains 83 sections, 406 equations, 10 figures.

Table of Contents

  1. Introduction and Summary
  2. Differentiation I: Basic Quadric Integrals
  3. The basic quadric integrals
  4. $E_{n,1}$
  5. $E_n$
  6. Divergence in the localization
  7. Case of degenerate $Q$
  8. The tangent configurations
  9. Summary of the general results
  10. Integrability of the symbols
  11. Scalar Diagrams: $n<d+2$
  12. Case of $n<d+2$
  13. E.g. 1: massive scalar box in 4d
  14. E.g. 2: massless scalar hexagon in 6d
  15. E.g. 3: pentagon in 4d
  16. ...and 68 more sections

Figures (10)

  • Figure 1: Geometric interpretation of $Q^{-1}L$.
  • Figure 2: Scalar $n$-gon.
  • Figure 3: Projection through the null space $\mathfrak{N}_Q$ of $Q$.
  • Figure 4: Spherical contour as a projection, viewed through the line $H_{(ij)}$.
  • Figure 5: Deformation of the ordinary residue contour into the new $S^1$ contour.
  • ...and 5 more figures