[Paper Review] A Fully Numerical Approach to One-Loop Amplitudes

M. Moretti, F. Piccinini+1 more|ArXiv.org|Feb 28, 2008

Particle physics theoretical and experimental studies31 references12 citations

TL;DR

This paper proposes a fully numerical approach to one-loop scattering amplitudes using dispersion relations to express loop amplitudes as convolutions of tree-level matrix elements, enabling efficient, automated computation with power-law scaling. The method avoids traditional tensor integral reductions and achieves accurate results by leveraging existing numerical tools for tree-level amplitudes, demonstrating feasibility and stability in the φ³ model.

ABSTRACT

We suggest a new approach for the automatic and fully numerical evaluation of one-loop scattering amplitudes in perturbative quantum field theory. We use suitably formulated dispersion relations to perform the calculation as a convolution of tree-level amplitudes. This allows to take advantage of the iterative numerical algorithms for the evaluation of leading order matrix elements.

Motivation & Objective

• To develop an automated, fully numerical method for computing one-loop scattering amplitudes in quantum field theory.
• To overcome the computational complexity and factorization scale dependence of leading-order calculations in multi-jet final states.
• To eliminate the need for analytical tensor integral reduction by reformulating loop amplitudes as convolutions of tree-level amplitudes.
• To enable stable and efficient numerical integration for NLO corrections in high-multiplicity processes.

Proposed method

• The method employs dispersion relations à la Veltman to express one-loop amplitudes as convolutions of tree-level matrix elements.
• It uses causal ordering and step functions (θ) to handle energy ordering in internal lines, ensuring correct analytic structure.
• The approach replaces traditional scalar and tensor integral reductions with numerical convolution integrals over phase space.
• The loop integration is performed numerically via Monte Carlo methods, combined with phase space integration.
• The method is formulated in Minkowski space with iε prescriptions to maintain causality and unitarity.
• It avoids the rational part problem in unitarity methods by directly computing the full amplitude through convolution.

Experimental results

Research questions

• RQ1 Can one-loop amplitudes be computed entirely numerically without analytical reduction of tensor integrals?
• RQ2 How can the computational complexity of NLO amplitudes for multi-jet final states be reduced to allow automation?
• RQ3 Can dispersion relations be used to express loop amplitudes as convolutions of tree-level amplitudes with controlled numerical stability?
• RQ4 How does the numerical convolution method compare to standard diagrammatic techniques in accuracy and efficiency?
• RQ5 Can infrared and collinear singularities be handled reliably in four-dimensional regularization within this numerical framework?

Key findings

• The method successfully computes one-loop amplitudes in the φ³ model using only tree-level matrix elements and numerical convolution.
• The approach achieves power-law scaling in computational complexity, avoiding the factorial growth typical of diagrammatic methods.
• Numerical results for the φ³ model show excellent agreement with standard analytical results, validating the method's accuracy.
• The method naturally handles causality and unitarity through the use of retarded propagators and step functions.
• The approach is stable under numerical integration and does not require explicit treatment of the rational part of the amplitude.
• The framework is fully automatable and compatible with existing Monte Carlo event generators based on tree-level amplitudes.