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On the computation of intersection numbers for twisted cocycles

Stefan Weinzierl

TL;DR

Weinzierl develops an extension-free algorithm for computing intersection numbers of twisted cocycles by combining a recursive variable-elimination strategy, pole-order reduction to simple poles, and a univariate global-residue evaluation. The framework relies on twisted cohomology and its dual, with a careful treatment of coefficient equivalence classes and gauge-like transformations to maintain invariance of the final result. The method is demonstrated through univariate and elliptic genus-1 examples, showing exact results without introducing algebraic extensions in intermediate steps, and it is applied to Feynman-integral contexts via the Baikov representation and maximal-cut techniques. The work provides practical tools for faster Feynman integral reductions and for deriving differential equations for master integrals in multivariate settings.

Abstract

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorithm, which avoids algebraic extensions.

On the computation of intersection numbers for twisted cocycles

TL;DR

Weinzierl develops an extension-free algorithm for computing intersection numbers of twisted cocycles by combining a recursive variable-elimination strategy, pole-order reduction to simple poles, and a univariate global-residue evaluation. The framework relies on twisted cohomology and its dual, with a careful treatment of coefficient equivalence classes and gauge-like transformations to maintain invariance of the final result. The method is demonstrated through univariate and elliptic genus-1 examples, showing exact results without introducing algebraic extensions in intermediate steps, and it is applied to Feynman-integral contexts via the Baikov representation and maximal-cut techniques. The work provides practical tools for faster Feynman integral reductions and for deriving differential equations for master integrals in multivariate settings.

Abstract

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorithm, which avoids algebraic extensions.

Paper Structure

This paper contains 21 sections, 8 theorems, 186 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. The recursive structure
  4. The equivalence class of the coefficients
  5. Reduction to simple poles
  6. The intersection number of univariate vector-valued one-forms with only simple poles
  7. Computation of the global residue
  8. Bases for the twisted cohomology groups
  9. The algorithm
  10. Examples
  11. An univariate example
  12. An example with an elliptic curve
  13. Third example
  14. Applications
  15. The Baikov representation
  16. ...and 6 more sections

Key Result

Proposition 1

Consider the cohomology class $\langle \varphi^{({\bf n})}_L | \in H^{({\bf n})}_\omega$ and expand $\langle \varphi^{({\bf n})}_L |$ in the basis of $H^{({\bf n-1})}_\omega$: Define $\hat{\varphi}^{({\bf n})}_{L,j}$ by $\varphi^{({\bf n})}_{L,j} = \hat{\varphi}^{({\bf n})}_{L,j} dz_n$. Changing the representative amounts to for some $(n-1)$-form $\xi$. Let us now consider transformations which

Figures (4)

  • Figure 1: The deformation of the contour. Singular points are drawn with a dot, critical points by a cross. Left: The original integration contour encircles all singular points anti-clockwise. Middle: The integration contour is deformed to infinity. Right: The integration contour is further deformed to enclose all critical points clockwise.
  • Figure 2: The Feynman graph corresponding to the two-loop sunrise integral. $p$ denotes the external momentum, $m_1$,$m_2$ and $m_3$ the internal masses.
  • Figure 3: Left: The integration contour in the unequal mass case with two singular points (dots) and one critical point (cross). Middle: The integration contour in the equal mass case with one singular points. Right: The one singular point can be considered as the limit where two singular points and one critical point coincide.
  • Figure 4: A non-planar Feynman diagram contributing to the mixed ${\mathcal{O}}(\alpha \alpha_s)$-corrections to the decay $H \rightarrow b \bar{b}$ through a $H t \bar{t}$-coupling. The Higgs boson is denoted by a dashed line, a top quark by a green line, a bottom quark with a black line and a gluon by a curly line. Particles with mass $m_W$ are drawn with a wavy line.

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 6 more