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

Heribertus Bayu Hartanto, Simon Badger+2 more|arXiv (Cornell University)|Jan 1, 2019

Particle physics theoretical and experimental studiesPhysics and Astronomy113 references56 citations

TL;DR

This paper presents the first numerical evaluation of planar two-loop helicity amplitudes for W-boson plus four partons in QCD, using finite-field sampling and integrand reduction to map amplitudes to master integrals. Despite the absence of a complete analytic master integral basis, the authors identify a numerically tractable set via local numerator insertions and achieve accurate results using sector decomposition, validating the universal two-loop pole structure in the HV scheme with high precision.

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.

Motivation & Objective

• To compute the first numerical results for two-loop helicity amplitudes in W-boson plus four parton scattering in QCD.
• To overcome the challenge of incomplete analytic master integral bases for multi-scale, off-shell amplitudes.
• To develop a numerically robust framework for evaluating amplitudes using finite-field sampling and integrand reduction.
• To validate the universal two-loop pole structure in the 't Hooft-Veltman scheme using numerical benchmark results.

Proposed method

• Employ finite-field sampling of Feynman diagrams to perform numerical integration-by-parts (IBP) reduction.
• Use local numerator insertions to identify a basis of master integrals with simplified divergence structures.
• Apply sector decomposition to numerically evaluate master integrals where analytic expressions are unavailable.
• Construct the amplitude via integrand reduction and IBP reduction over finite fields, avoiding large symbolic IBP tables.
• Validate results by comparing divergent parts against the universal two-loop pole structure in the HV scheme.
• Cross-check results using a diagram-based integrand reduction approach with direct evaluation via Fiesta/pySecDec.

Experimental results

Research questions

• RQ1 Can two-loop helicity amplitudes for W+4 partons be evaluated numerically despite the lack of a complete analytic master integral basis?
• RQ2 How can a numerically stable basis of master integrals be identified for amplitudes with an off-shell vector boson?
• RQ3 To what extent can the universal two-loop pole structure be recovered numerically in the 't Hooft-Veltman scheme?
• RQ4 Can finite-field sampling and integrand reduction provide reliable numerical results without full analytic IBP tables?
• RQ5 How accurate is the numerical evaluation of the amplitude when compared to the expected universal pole structure?

Key findings

• The numerical evaluation of the two-loop amplitude for q̄Q Q̄q′νℓ and qggq′νℓ processes agrees with the universal two-loop pole structure to O(ϵ⁻²) with high precision.
• The divergent part of the amplitude matches the universal pole structure P(2) defined in Eq. (3.6), with uncertainties assessed consistently at the O(ϵ⁻¹) level.
• The numerical results for the finite remainder (after UV and IR subtraction) are consistent across independent evaluation methods, including diagram-based integrand reduction.
• The O(ϵ⁻¹) pole coefficients in the q̄Q Q̄q′νℓ channel are reproduced with a relative uncertainty of about 10⁻⁵, confirming numerical stability.
• The results for the qggq′νℓ channel show agreement with the universal pole structure to within 0.1% for the O(ϵ⁻²) term and better than 0.2% for O(ϵ⁻¹) terms.
• The use of local numerator insertions successfully identifies a numerically stable master integral basis, enabling accurate sector decomposition evaluation.