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A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons

Heribertus Bayu Hartanto, Simon Badger, Christian Brønnum-Hansen, Tiziano Peraro

TL;DR

This work tackles the challenge of computing planar two-loop helicity amplitudes for a $W$-boson produced with four partons in QCD, a multi-scale, multi-leg problem relevant for NNLO predictions. The authors implement a fully numerical pipeline based on finite-field sampling to reduce Feynman diagrams to master integrals via IBP, complemented by integrand reduction and a local-numerator master-integral basis to stabilize divergent integrals; sector decomposition then provides numerical evaluations for the master integrals. They deliver the first numerical results for these planar two-loop amplitudes, validating the pole structure against universal IR/UV predictions and cross-checking with an independent integrand-reduction approach. This demonstrates the viability of a finite-field, numerically reconstructed strategy for complex multi-leg two-loop amplitudes and establishes a foundation for analytic reconstruction and improved phenomenological predictions for $pp\to W$+jets.

Abstract

We present the first numerical results for the two-loop helicity amplitudes for the scattering of four partons and a W-boson in QCD. We use a finite field sampling method to reduce directly from Feynman diagrams to the coefficients of a set of master integrals after applying integration-by-parts identities. Since the basis of master integrals is not yet fully known analytically, we identify a set of master integrals with a simple divergence structure using local numerator insertions. This allows for accurate numerical evaluation of the amplitude using sector decomposition methods.

A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons

TL;DR

This work tackles the challenge of computing planar two-loop helicity amplitudes for a -boson produced with four partons in QCD, a multi-scale, multi-leg problem relevant for NNLO predictions. The authors implement a fully numerical pipeline based on finite-field sampling to reduce Feynman diagrams to master integrals via IBP, complemented by integrand reduction and a local-numerator master-integral basis to stabilize divergent integrals; sector decomposition then provides numerical evaluations for the master integrals. They deliver the first numerical results for these planar two-loop amplitudes, validating the pole structure against universal IR/UV predictions and cross-checking with an independent integrand-reduction approach. This demonstrates the viability of a finite-field, numerically reconstructed strategy for complex multi-leg two-loop amplitudes and establishes a foundation for analytic reconstruction and improved phenomenological predictions for +jets.

Abstract

We present the first numerical results for the two-loop helicity amplitudes for the scattering of four partons and a W-boson in QCD. We use a finite field sampling method to reduce directly from Feynman diagrams to the coefficients of a set of master integrals after applying integration-by-parts identities. Since the basis of master integrals is not yet fully known analytically, we identify a set of master integrals with a simple divergence structure using local numerator insertions. This allows for accurate numerical evaluation of the amplitude using sector decomposition methods.

Paper Structure

This paper contains 7 sections, 33 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Calculational framework
  3. Planar two-loop $W$ plus four parton scattering
  4. Local master integrals
  5. Numerical results
  6. Conclusions
  7. Kinematic invariants at the benchmark phase space point

Figures (3)

  • Figure 1: Independent maximal cut topologies contributing to planar $W+4$ parton scattering at two-loops. The full set of 15 maximal cuts can be obtained by including 2 permutations of $A_1$, $A_3$, $B_1$, $B_2$, $C_1$ and $C_2$ topologies.
  • Figure 2: Master integrals for leading colour $W+4$ parton scattering at two loops with five external legs. $(a,b)$ represents the number of crossing of external legs ($a$) and the number master integral for a given topology ($b$). A massless (massive) external leg is indicated by a single (double) line external leg. The $\ast$ sign identifies master integral topologies that are not known analytically.
  • Figure 3: Master integrals for leading colour $W+4$ parton scattering at two loops with four external legs or fewer. $(a,b)$ represents the number of crossing of external legs ($a$) and the number master integral for a given topology ($b$). A massless (massive) external leg is indicated by a single (double) line external leg. All master integral topologies shown are known analytically.