[Paper Review] Automation of 2-loop Amplitude Calculations

Stephen Jones|arXiv (Cornell University)|Aug 12, 2016

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper presents an automated framework for computing two-loop amplitudes in quantum field theory, focusing on Higgs boson pair production at NLO in QCD. It combines sector decomposition with Quasi-Monte Carlo (QMC) integration to numerically evaluate master integrals, enabling the first fully numerical computation of the differential cross-section for $gg \to HH$ using a GPU-accelerated, phase-space sampling-optimized workflow.

ABSTRACT

Some of the tools and techniques that have recently been used to compute Higgs boson pair production at NLO in QCD are discussed. The calculation relies on the use of integral reduction, to reduce the number of integrals which must be computed, and expressing the amplitude in terms of a quasi-finite basis, which simplifies their numeric evaluation. Emphasis is placed on sector decomposition and Quasi-Monte Carlo (QMC) integration which are used to numerically compute the master integrals.

Motivation & Objective

• Develop an automated, numerically robust framework for precision calculations beyond NLO in QCD.
• Address the challenge of computing 2-loop amplitudes with multiple scales when analytical results are unavailable.
• Minimize the number of phase-space points required for accurate cross-section evaluation using optimized sampling techniques.
• Enable efficient numerical evaluation of master integrals via sector decomposition and QMC integration.
• Leverage GPU acceleration and multi-threading to achieve high-performance computation of virtual amplitudes.

Proposed method

• Use conventional dimensional regularization (CDR) with $D = 4 - 2\epsilon$ to handle ultraviolet and infrared divergences.
• Decompose the amplitude into form factors $F_1, F_2$ contracted with projectors $P_1^{\mu\nu}, P_2^{\mu\nu}$ to isolate loop integrals.
• Apply integral reduction via REDUZE to reduce the number of independent integrals, though non-planar 4-point integrals required alternative treatment.
• For integrals not amenable to analytical reduction, apply sector decomposition to isolate singularities and prepare for numerical integration.
• Implement Quasi-Monte Carlo (QMC) integration using the R1SL and VEGAS algorithms to evaluate decomposed integrals with high precision.
• Use OpenCL to parallelize the numerical integration across CPUs and GPUs, achieving high throughput and linear scaling with core count.

Experimental results

Research questions

• RQ1 How can 2-loop amplitudes with multiple scales be computed numerically when analytical solutions are unavailable?
• RQ2 What is the performance and accuracy of QMC integration when applied to sector-decomposed, quasi-finite integrals in a high-dimensional phase space?
• RQ3 To what extent can phase-space sampling be optimized to reduce the number of required amplitude evaluations without compromising precision?
• RQ4 How effectively can GPU-accelerated, multi-threaded implementations scale for numerically intensive 2-loop amplitude computations?
• RQ5 Can a fully automated framework be built that integrates integral reduction, sector decomposition, and numerical integration for generic 2-loop processes?

Key findings

• The first numerical computation of the differential cross-section for Higgs boson pair production ($gg \to HH$) at NLO in QCD was achieved using this framework.
• The total cross-section was computed with a statistical uncertainty of 0.3% from event sampling and an additional 0.1% uncertainty from numerical integration of master integrals.
• The median GPU time per phase-space point was 2 hours, with a total of 4680 GPU hours used over 6 days of wall-clock time.
• The implementation achieved linear scaling with CPU core count up to physical core limits, with significant speed-up using a single Nvidia Tesla K20Xm GPU.
• VEGAS and R1SL QMC algorithms both performed well, with R1SL showing better scaling for larger function dimensions, and VEGAS delivering near-optimal Monte Carlo scaling.
• The framework successfully computed 666 phase-space points without using phase-space grids, enabling flexible re-evaluation for different cuts, PDFs, or center-of-mass energies.