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Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

Alessandro Georgoudis, Yang Zhang

Abstract

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.

Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

Abstract

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.

Paper Structure

This paper contains 13 sections, 4 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Integral Reduction via the Analysis of Algebraic Curves
  3. Elliptic Example: Double Box with Internal Masses
  4. Maximal unitarity
  5. Integral reduction
  6. Reduction of the double-propagator integrals
  7. Elliptic Example: Sunset Diagram
  8. Hyperelliptic Example: Nonplanar Crossed Box with Internal Masses
  9. Maximal Unitarity and geometric properties
  10. Integral reduction
  11. Conclusions
  12. Rudiments of Algebraic Curves
  13. Riemann-Roch theorem

Key Result

Theorem 1

(Riemann-Hurwitz) Let $f : X\rightarrow S$ be a meromorphic function of degree $d$ on a closed connected Riemann surface $X$. The ramified points are $x_1,\ldots, x_n$, with multiplicity $m_1,\ldots, m_n$ . Then the Euler character of $X$,

Figures (3)

  • Figure 1: Planar double box diagram with $3$ internal mass scales
  • Figure 2: Sunset diagram
  • Figure 3: Nonplanar double box

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Theorem 2
  • Theorem 3
  • Corollary 1