Scattering Amplitudes with Open Loops (1111.5206v2)

Abstract: We introduce a new technique to generate scattering amplitudes at one loop. Traditional tree algorithms, which handle diagrams with fixed momenta, are promoted to generators of loop-momentum polynomials that we call open loops. Combining open loops with tensor-integral and OPP reduction results in a fully flexible, very fast, and numerically stable one-loop generator. As demonstrated with non-trivial applications, the open-loop approach will permit to obtain precise predictions for a very wide range of collider processes.

• The paper introduces an innovative open loop concept that extends tree-level algorithms to compute one-loop scattering amplitudes efficiently.
• The approach reduces algebraic complexity and speeds up numerical evaluations, achieving sub-second runtimes for complex collider processes.
• The method’s flexibility in combining tensor-integral and OPP reduction paves the way for robust simulations of multi-particle interactions.

Overview of the Paper on Scattering Amplitudes with Open Loops

The paper introduces a novel method for generating one-loop scattering amplitudes, termed "open loops." This technique harnesses traditional tree algorithms to create loop-momentum polynomials, transforming these into open loops which are then paired with tensor-integral and OPP reduction. This combined approach results in a highly adaptable, rapid, and numerically stable generator for one-loop computations, with significant applications in predicting outcomes of collider processes, particularly those relevant to the Large Hadron Collider (LHC).

Key Contributions

The authors outline several contributions to the field:

1. Open Loop Concept: By extending tree-level algorithms, they introduce a method for computing scattering amplitudes at the one-loop level using open loops. This approach not only facilitates the calculation of complex processes but also handles loop-momentum dependencies efficiently.
2. Efficient Numerical Evaluation: The open loop method enhances computational efficiency by reducing the required computational resources for evaluating scalar-integral coefficients through the OPP framework and tensor-integral methods.
3. Reduction of Algebraic Complexity: The technique mitigates the algebraic complexity associated with traditional one-loop methods, leading to manageable code and resource requirements even for intricate processes with high particle multiplicity.

Numerical Results and Performance

The results demonstrate that the new approach is both fast and precise. Runtime analysis for various collider processes shows nearly linear scaling of computational time with the number of diagrams—a significant advancement in the context of high-energy physics simulations. Notably, for $2 \to 4$ processes at the LHC energy scales, the runtime per phase-space point is considerably reduced compared to traditional methods, often staying below one second per evaluation.

Implications and Future Directions

This work has substantial implications for theoretical and practical aspects of particle physics research. It allows researchers to efficiently simulate next-to-leading-order (NLO) processes, which are essential for accurately interpreting collider data and testing the validity of theoretical models. The flexibility of open loops, which can be combined with varying reduction techniques, enhances their applicability across a broader spectrum of high-energy physics problems.

In terms of future developments, the authors indicate the potential for their method to efficiently handle complex multi-particle processes, which are anticipated to become increasingly relevant as collider experiments continue to evolve. Furthermore, potential improvements in numerical stability, particularly when using OPP reduction, are areas for future exploration. Techniques like incorporating quadruple precision or employing interpolation for numerically unstable points could provide further advances in computational reliability and accuracy.

This paper represents a significant step in optimizing one-loop amplitude calculations, offering a robust, scalable solution to challenges faced in simulating the high-energy interactions at modern colliders. It lays the groundwork for further advancements in both theoretical insights and practical simulations within the field of particle physics.