Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards numerical two-loop integrand reduction

Giuseppe Bevilacqua, Dhimiter Canko, Costas Papadopoulos, Aris Spourdalakis

TL;DR

This work develops a general framework for integrand-level reduction of two-loop helicity amplitudes in $d=4-2\epsilon$ and $d=4$, expressing the amplitude as ${\cal A}^{(2)}=\sum_i c_i(\{p\})\,F_i$ with integrals ${F_i}$ and an ISP-based numerator expansion ${\cal N}=P^{(n)}+\sum P_i^{(n-1)}D_i+\cdots$. It introduces two complementary coefficient-extraction strategies: a linear-fit, cut-based approach and a global-fit approach, and further proposes projecting onto a full integral family to reduce the polynomial complexity. The method is tested across 4-, 5-, and 6-point processes, including double-box, penta-box, penta-triangle, hexabubble, non-planar double-box, $t\bar t$, $t\bar t H$, and six-gluon topologies, with both analytic and numerically supplied numerators, demonstrating consistent reconstruction of integrand coefficients. Findings indicate that projecting over a full family and using the two-fit strategies yields a scalable path toward automated two-loop amplitude calculations for arbitrary processes, compatible with IBP-based master integral reductions. The work lays the groundwork for integrating this reduction with numerical numerator evaluation (via HELAC-2LOOP) and existing master integral libraries, advancing toward a fully automated two-loop computational framework. The results hold potential to significantly improve precision predictions at NNLO and beyond for multi-leg scattering processes at current and future colliders.

Abstract

We present a method for the integrand-level reduction of two-loop helicity amplitudes in both $d=4-2ε$ and $d=4$ dimensions. The amplitude is expressed in terms of a set of Feynman integrals and their coefficients that depend on the external kinematics. The analysis presented in this paper, in conjunction with the ongoing development of the computational framework $\text{HELAC-2LOOP}$, paves the road for the construction of an automated program for two-loop amplitude calculations for arbitrary scattering processes.

Towards numerical two-loop integrand reduction

TL;DR

This work develops a general framework for integrand-level reduction of two-loop helicity amplitudes in and , expressing the amplitude as with integrals and an ISP-based numerator expansion . It introduces two complementary coefficient-extraction strategies: a linear-fit, cut-based approach and a global-fit approach, and further proposes projecting onto a full integral family to reduce the polynomial complexity. The method is tested across 4-, 5-, and 6-point processes, including double-box, penta-box, penta-triangle, hexabubble, non-planar double-box, , , and six-gluon topologies, with both analytic and numerically supplied numerators, demonstrating consistent reconstruction of integrand coefficients. Findings indicate that projecting over a full family and using the two-fit strategies yields a scalable path toward automated two-loop amplitude calculations for arbitrary processes, compatible with IBP-based master integral reductions. The work lays the groundwork for integrating this reduction with numerical numerator evaluation (via HELAC-2LOOP) and existing master integral libraries, advancing toward a fully automated two-loop computational framework. The results hold potential to significantly improve precision predictions at NNLO and beyond for multi-leg scattering processes at current and future colliders.

Abstract

We present a method for the integrand-level reduction of two-loop helicity amplitudes in both and dimensions. The amplitude is expressed in terms of a set of Feynman integrals and their coefficients that depend on the external kinematics. The analysis presented in this paper, in conjunction with the ongoing development of the computational framework , paves the road for the construction of an automated program for two-loop amplitude calculations for arbitrary scattering processes.

Paper Structure

This paper contains 19 sections, 51 equations, 8 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Description of the method
  3. Basic notation and conventions
  4. Linear fit and fit by cut approach
  5. Global fit approach
  6. Projecting over a full family
  7. Application of the method
  8. Four-Point Kinematics
  9. Double-box topologies
  10. Penta-triangle topologies
  11. Hexa-bubble topologies
  12. Non-planar double box topologies
  13. $gg\to t\bar{t}$
  14. Five-Point Kinematics
  15. Penta-box topologies
  16. ...and 4 more sections

Figures (8)

  • Figure 1: Feynman graphs contributing to the double-box numerator under study. There are seven contributions, considering gluons and (anti-)ghosts running within the loops. Curly lines denote gluons, while dotted lines denote ghosts (arrow aligned with the momentum flow) and anti-ghosts (arrow in the opposite direction of the momentum flow).
  • Figure 2: Feynman graphs contributing to the penta-triangle numerator under study. There are seven contributions, considering gluons and (anti-)ghosts running within the loops.
  • Figure 3: Feynman graphs contributing to the hexa-bubble numerator under study. There are four contributions, considering gluons and (anti-)ghosts running within the loops.
  • Figure 4: Feynman graphs contributing to the non-planar double-box numerator under study. There are four contributions, considering gluons and (anti-)ghosts running within the loops.
  • Figure 5: Feynman graphs contributing to the $gg\to t\bar{t}$ numerator under study. There are three contributions, considering gluons and (anti-)ghosts running within the $k_1$ loop, which are collected in the first line of this Figure. In the second line, we quote the tree-order graph used for the summation over polarizations. The thick black line indicates the (anti-)top quark.
  • ...and 3 more figures