Numerical Integrator Advances Precision of Multi-Loop Feynman Integral Evaluation

Calculating Feynman integrals, essential components of particle physics calculations, often presents a major computational bottleneck, hindering precise predictions for experiments like those at the Large Hadron Collider. Pau Petit Rosasa and William J. Torres Bobadilla, both from the Department of Mathematical Sciences at the University of Liverpool, and their colleagues have developed a new method to dramatically speed up these calculations. Their approach numerically solves equations that Feynman integrals satisfy, allowing them to handle the complex mathematical challenges posed by these integrals with unprecedented efficiency. This advancement enables real-time calculation of Feynman integrals within Monte Carlo generators, and facilitates the efficient creation of complex datasets previously considered too computationally expensive, promising to accelerate the development of more accurate and detailed particle physics simulations.

High-Precision Calculations of Scattering Amplitudes

The pursuit of increasingly precise theoretical predictions is fundamental to modern particle physics, allowing scientists to rigorously compare theory with experimental observations. These predictions rely on calculating scattering amplitudes, but calculating these at high precision, specifically beyond next-to-leading order, presents a significant computational challenge due to the complexity of multi-loop Feynman integrals. These integrals become particularly difficult when dealing with processes involving many particles or massive internal states, pushing current computational techniques to their limits. A standard approach involves using integration-by-parts identities to reduce Feynman integrals to a manageable set of master integrals, which then satisfy systems of differential equations.

Existing methods often rely on power series expansions or numerical integration of these equations, but these tools often struggle to integrate seamlessly into high-performance Monte Carlo event generators, essential for simulating particle collisions. Current limitations stem from the need for pre-computed grids, which become impractical as the number of variables increases, and the challenges of efficiently evaluating the algebraic functions that appear in the differential equations. A fully numerical integration approach offers a potential solution by bypassing the need for these grids and enabling on-the-fly evaluation of integrals, but this has been largely unexplored for complex, high-dimensional scattering processes. This research addresses these challenges by presenting a new C++ framework designed for the rapid numerical evaluation of systems of differential equations.

The framework is engineered to interface directly with Monte Carlo event generators, eliminating the need for pre-computed grids and enabling calculations for processes with a large number of variables. A key innovation lies in the treatment of algebraic functions, allowing for robust and stable numerical integration by studying and continuing their analytic structure. By expressing scattering amplitudes in terms of simplified transcendental functions, the researchers have streamlined the analytic structure and facilitated the numerical evaluation of the underlying differential equations.

Numerical Solution Accelerates Feynman Integral Calculation

Researchers have developed a new computational framework for calculating Feynman integrals, essential for precise theoretical predictions in particle physics. These integrals become extraordinarily difficult to calculate at high levels of accuracy, particularly when many particles are involved. Current methods often rely on pre-computed grids or complex series expansions, limiting their efficiency and scalability. This new approach employs a fully numerical integration technique, directly solving the differential equations that define these integrals without relying on pre-calculation. This is a significant departure from existing tools and overcomes limitations encountered when dealing with complex scenarios and a large number of variables.

The system is implemented in C++, designed for seamless integration with Monte Carlo event generators, and eliminates the need for impractical pre-computed grids, allowing for “on-the-fly” calculation of integrals. The integrator demonstrates a marked performance advantage over existing methods, achieving millisecond-level execution times for one-loop calculations and hundreds of milliseconds for more complex two-loop calculations. This speed allows for the efficient generation of grids when high precision is required, and the system is readily parallelizable, further reducing computation time. A key innovation lies in the method’s ability to handle the complex algebraic functions within these equations, ensuring stable and robust numerical integration across different physical regions.

The researchers have successfully applied this framework to several challenging calculations, including processes involving the creation of multiple particles and those with internal massive particles. This advancement promises to significantly accelerate the development of next-generation theoretical predictions, enabling more precise comparisons between theoretical models and experimental observations at the forefront of particle physics research. The ability to perform these calculations efficiently will be crucial for interpreting data from current and future collider experiments.

Numerical Solution of Multi-Loop Feynman Integrals

Researchers have developed a novel computational strategy for tackling the complex calculations required in particle physics, specifically the evaluation of Feynman integrals which underpin predictions for collider experiments. These integrals become increasingly difficult to solve as the complexity of the interaction increases, creating a bottleneck in achieving highly accurate theoretical predictions. This method distinguishes itself through a fully numerical integration technique, avoiding the limitations of approaches that depend on finding analytical solutions or power series expansions. The researchers built an integrator capable of evaluating integrals with both standard and extended precision, significantly reducing computation times compared to existing tools.

A key innovation lies in how the integrator handles inherent complexities within the integrals, ensuring accurate and stable results. To further enhance performance, the integrator incorporates strategies for optimising the integration path and minimising computational demands. This includes careful error control and numerical stabilisation techniques, ensuring the reliability of the results even for highly complex integrals. The method was successfully tested on several challenging scenarios, including calculations relevant to the production of top quarks and associated jets, and the radiative return process in electron-positron annihilation.

The resulting integrator demonstrates remarkable speed, achieving millisecond-level performance for one-loop calculations and hundreds of milliseconds for more complex two-loop scenarios. This speed is crucial for integrating the calculations into Monte Carlo event generators, essential for simulating particle collisions and comparing theoretical predictions with experimental data, and for efficiently generating the necessary data for complex calculations. This advancement promises to accelerate the development of more accurate theoretical models and facilitate more precise tests of fundamental physics.

Fast Numerical Integration of Multi-Loop Integrals

This work presents a novel numerical integrator for evaluating multi-loop Feynman integrals, a crucial step in achieving high-precision theoretical predictions for particle physics. The method centres on directly solving the differential equations that govern these integrals, with a particular focus on efficiently handling the inherent complexities in their mathematical structure. Demonstrating its capabilities, the integrator successfully evaluated complex one- and two-loop integrals, including those relevant to specific particle production processes and electron-positron annihilation, achieving execution times ranging from milliseconds to hundreds of milliseconds. The resulting speed improvements represent a significant advance, potentially enabling the incorporation of these calculations into Monte Carlo event generators and facilitating the efficient creation of grids for complex topologies that were previously computationally prohibitive. The authors acknowledge that the current implementation, while effective, is limited by its computational architecture for integration into high-performance generators. Future work will focus on addressing these limitations and further optimising the integrator for broader application.