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Calculating gluon one-loop amplitudes numerically

Jan-Christopher Winter, Walter T. Giele

TL;DR

The paper presents a semi-numerical, C++ implementation of one-loop gluon amplitudes using generalized unitarity, following established formalism to decompose amplitudes into cut-constructible and rational parts with $D_s$-dimension handling. It reconstructs coefficients of box, triangle, and bubble integrals from M-point unitarity cuts, using tree-level amplitudes (via Berends–Giele recursion) as inputs and solving for residues. Validation against analytic results confirms correct gauge behavior and pole structures, while studies reveal accurate results up to moderate $N$ in double precision and reveal precision limits that motivate higher-precision arithmetic for larger $N$. Efficiency analyses show polynomial scaling consistent with expectations (roughly $N^9$) and demonstrate practical running times on standard hardware. The work demonstrates a viable, integrable C++ tool for one-loop gluon amplitudes that can be embedded in future NLO Monte Carlo generators.

Abstract

This note reports on an independent implementation of calculating one-loop amplitudes semi-numerically using generalized unitarity techniques. The algorithm implemented in form of a C++ code closely follows the method by Ellis, Giele, Kunszt and Melnikov. For the case of gluons, the algorithm is briefly reviewed. Double-precision results are presented documenting the accuracy and efficiency of this computation.

Calculating gluon one-loop amplitudes numerically

TL;DR

The paper presents a semi-numerical, C++ implementation of one-loop gluon amplitudes using generalized unitarity, following established formalism to decompose amplitudes into cut-constructible and rational parts with -dimension handling. It reconstructs coefficients of box, triangle, and bubble integrals from M-point unitarity cuts, using tree-level amplitudes (via Berends–Giele recursion) as inputs and solving for residues. Validation against analytic results confirms correct gauge behavior and pole structures, while studies reveal accurate results up to moderate in double precision and reveal precision limits that motivate higher-precision arithmetic for larger . Efficiency analyses show polynomial scaling consistent with expectations (roughly ) and demonstrate practical running times on standard hardware. The work demonstrates a viable, integrable C++ tool for one-loop gluon amplitudes that can be embedded in future NLO Monte Carlo generators.

Abstract

This note reports on an independent implementation of calculating one-loop amplitudes semi-numerically using generalized unitarity techniques. The algorithm implemented in form of a C++ code closely follows the method by Ellis, Giele, Kunszt and Melnikov. For the case of gluons, the algorithm is briefly reviewed. Double-precision results are presented documenting the accuracy and efficiency of this computation.

Paper Structure

This paper contains 4 sections, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Construct one-loop by tree-level amplitudes -- algorithm in brief
  3. C++ implementation and results
  4. Conclusions

Figures (3)

  • Figure 1: Double-, single-pole and finite-part accuracy (in double precision) of the $++--\ldots$ one-loop amplitudes for $N=6,\ldots,11,15$ gluons; see also text and right panel of Figure \ref{['Fig:speed']}. Bottom row: center, double-logarithmic distributions of Gram determinants involving sets of external gluons and, right, $e^{(0)}_{i_1\cdots i_5}$ coefficients for the $N=6,8,15$ gluon setups.
  • Figure 2: Finite-part accuracy versus minimal Gram determinant of external-gluon sets (left), maximal $e^{(0)}_{i_1\cdots i_5}$ coefficient (center) and single-pole accuracy (all in double precision) for the $N=6,8,11$ gluon setups of above.
  • Figure 3: $N$ dependence of the computing time; $x_N$-exponents (center), see text. Times refer to using a 2.20 GHz Intel Core2 Duo processor. Tighter gluon cuts were used: $|\eta_i|<2$, $p_{\perp,i}\,s^{-0.5}>0.1$, $R_{ij}>0.7$, denoted as in Giele:2008bc. The last plot shows the accuracy improvement.