[Paper Review] Multi-gluon one-loop amplitudes numerically

Achilleas Lazopoulos|ArXiv.org|Dec 16, 2008

Particle physics theoretical and experimental studies22 references24 citations

TL;DR

This paper presents a C++ implementation of the $D_s$-dimensional unitarity cut algorithm for numerically computing one-loop multi-gluon amplitudes in NLO QCD, using tree-level color-ordered amplitudes with gluons and scalars in five dimensions as building blocks. The method enables accurate evaluation of both the rational and cut-constructible parts of virtual amplitudes, with numerical stability ensured via quadruple-precision fallback, achieving full agreement with published results for up to 22 external gluons.

ABSTRACT

A c++ implementation of the D_s-dimensional unitarity cut algorithm for the numerical evaluation of the virtual contribution to NLO QCD amplitudes is presented. The current version includes an arbitrary number of external gluons with gluonic propagators in the loop. The building blocks are tree level color-ordered amplitudes with gluons and with gluons and two scalars in five dimensions. Numerical stability issues are addressed and agreement has been reached with the results published in the literature.

Motivation & Objective

• To develop a generic, numerically stable C++ tool for computing one-loop virtual amplitudes in NLO QCD with arbitrary numbers of external gluons.
• To implement the $D_s$-dimensional unitarity cut method (EGKM algorithm) for efficient and accurate evaluation of both rational and cut-constructible parts of amplitudes.
• To ensure numerical robustness in high-multiplicity processes by integrating quadruple-precision arithmetic as a fallback mechanism.
• To enable future extension to fermions and gauge bosons within the same framework, supporting phenomenologically relevant processes at the LHC.

Proposed method

• The algorithm uses the OPP reduction method at the integrand level, decomposing one-loop amplitudes into scalar box, triangle, bubble, and tadpole integrals.
• The virtual amplitude is expressed as $A^{D_s} = A^0 + A^1 \cdot D_s$, where $D_s$ is the dimensionality of internal particles, enabling numerical evaluation at two $D_s$ values to extract both the $\epsilon$-dependent and rational parts.
• Tree-level color-ordered amplitudes with gluons and gluons plus two scalars in five dimensions are computed using the Berends-Giele recursion relation.
• The method employs complex unitarity cuts in $D_s$ dimensions to reconstruct the amplitude, with numerical stability maintained via a quadruple-precision version of the code using the QD library.
• The algorithm is applied to N-gluon amplitudes with arbitrary external helicities and phase-space points, with performance measured via average CPU time over $10^5$ phase-space points.
• Checks are performed against published results for specific phase-space points, confirming agreement within numerical precision.

Experimental results

Research questions

• RQ1 How can one numerically compute the full virtual one-loop amplitude, including the rational part, for multi-gluon processes in NLO QCD?
• RQ2 What numerical strategies are effective in maintaining stability for high-multiplicity amplitudes with many external particles?
• RQ3 To what extent can the $D_s$-dimensional unitarity cut method be implemented efficiently and robustly in a generic C++ framework?
• RQ4 How does the performance of the algorithm scale with the number of external gluons, and what computational resources are required for phenomenologically relevant processes?

Key findings

• The C++ implementation successfully computes full one-loop amplitudes for up to 22 external gluons, with typical CPU times ranging from 3.4 ms for 4 gluons to 1.85 seconds for 22 gluons on an Intel Xeon X5450.
• For $N=6$ gluons, approximately 5% of phase-space points required quadruple-precision arithmetic when the switch accuracy was set to $10^{-4}$, indicating manageable performance overhead.
• The relative error in double and single poles for $N=6$ showed a logarithmic distribution, with peaks shifting toward larger errors as $N$ increased, but the finite part remained stable in most cases.
• The code achieves full agreement with published results from the literature, including checks at $D_s = 5$ and $6$, confirming correctness of the implementation.
• The method correctly recovers both the $1/\epsilon^2$ and $1/\epsilon$ poles, as well as the rational finite part, in the $\epsilon$-expansion of the amplitude.
• The framework is extensible to fermions and gauge bosons, as demonstrated in prior work, and the path to full phenomenological implementation is now established.