An Automated Implementation of On-Shell Methods for One-Loop Amplitudes (0803.4180v1)

Abstract: We present the first results from BlackHat, an automated C++ program for calculating one-loop amplitudes. The program implements the unitarity method and on-shell recursion to construct amplitudes. As input to the calculation, it uses compact analytic formulae for tree amplitudes for four-dimensional helicity states. The program performs all related computations numerically. We make use of recently developed on-shell methods for evaluating coefficients of loop integrals, introducing a discrete Fourier projection as a means of improving efficiency and numerical stability. We illustrate the numerical stability of our approach by computing and analyzing six-, seven- and eight-gluon amplitudes in QCD and comparing against previously-obtained analytic results.

• The paper introduces an automated C++ implementation using on-shell methods to compute one-loop amplitudes in QCD.
• It employs generalized unitarity techniques and on-shell recursion to separate cut contributions from rational terms while ensuring numerical stability.
• The method efficiently scales to multi-gluon processes and is pivotal for advancing NLO calculations in collider physics.

Overview of Automated On-Shell Methods for One-Loop Amplitudes

The paper "An Automated Implementation of On-Shell Methods for One-Loop Amplitudes" presents a computational approach for evaluating one-loop amplitudes in quantum chromodynamics (QCD) using on-shell methods. The authors introduce an automated C++ program designed to efficiently and accurately compute these amplitudes by leveraging the unitarity method and on-shell recursion. This approach is particularly significant in the context of collider physics, such as the Large Hadron Collider (LHC), where precise theoretical predictions for Standard Model processes are crucial for identifying potential signals of new physics.

Methodology

The methodology employed in this program revolves around the decomposition of one-loop amplitudes into two main components: cut contributions and rational terms. The cut contributions are derived from scalar basis integrals, including box, triangle, and bubble integrals, which are calculated using the four-dimensional unitarity method. This calculation is expedited by the use of compact analytic tree-level amplitudes.

A notable aspect of the approach is the application of generalized unitarity techniques, which involve evaluating coefficients of integral functions through discrete Fourier projections. This enables numerical stability and computational efficiency, minimizing reliance on traditional Feynman-diagrammatic methods.

For the rational terms, the program uses on-shell recursion relations analogous to tree-level recursion but adapted for loop-level amplitudes. This aspect is crucial in reconstructing the rational terms that do not emerge from cut integrals but from the loop momenta's off-shell behavior.

Numerical Stability and Testing

The authors address the inherent numerical challenges in computing loop amplitudes by implementing checks for stability at phase-space integration points. This includes ensuring the cancellation of spurious singularities and correctly reproducing the known infrared ($1/\epsilon$) singularities associated with bubble integrals. In instances where numerical instability is detected, the program recomputes the amplitudes at higher precision using the QD library, which supports quad-double arithmetic.

The program's effectiveness is demonstrated through evaluations of six-, seven-, and eight-gluon amplitudes. These amplitudes are compared against known analytic results and show excellent agreement, confirming both the accuracy and stability of the implemented methodology.

Computational Efficiency

The computational demands of the method are shown to scale well with increasing numbers of external particles, a crucial requirement for practical use in LHC phenomenology. Despite the complexity of the methods involved, the program achieves notable efficiency, completing calculations for six-gluon amplitudes in an average time of under 120 milliseconds per phase-space point even when using multi-precision for stability.

Implications and Future Directions

The automated program signifies a step forward in the computational capability for one-loop amplitudes, offering a tool that combines efficiency and numerical stability, suitable for integrating with higher-level NLO calculations. This development is critical for advancing the accuracy of theoretical predictions required for analyzing collider data, potentially revealing new physics beyond the Standard Model.

Moving forward, the frameworks and algorithms presented in this paper will likely extend to processes involving external fermions and massive particles, further enhancing their applicability. As the authors propose, a broader implementation of these methods could become publicly available, supporting a wider community in theoretical and phenomenological studies in particle physics.