- The paper introduces Kira, a tool that efficiently applies a modular arithmetic-enhanced Laporta algorithm for reducing Feynman integrals.
- It leverages integration-by-parts and Lorentz invariance identities with advanced algebra systems to simplify multi-loop integrals into master forms.
- Kira demonstrates competitive performance in NNLO collider benchmarks, reducing computational overhead compared to existing reduction programs.
Overview of "Kira -- A Feynman Integral Reduction Program"
The paper "Kira -- A Feynman Integral Reduction Program" authored by P. Maierhöfer, J. Usovitsch, and P. Uwer introduces a new implementation of the Laporta algorithm for the reduction of Feynman integrals. The program, named Kira, is designed to efficiently handle scalar multi-loop integrals commonly found in quantum field theoretic calculations, notably in high precision collider experiments. These complex multi-loop integrals need to be expressed as a sum of simpler master integrals, facilitating their computation and application in next-to-next-to-leading order (NNLO) perturbative expansions.
Methodology and Implementation
Kira distinguishes itself from previous implementations through the integration of modular arithmetic to remove linearly dependent equations from those generated by integration-by-parts (IBP) and Lorentz invariance (LI) identities. This modular approach minimizes excessive algebraic manipulations by transforming complex rational functions into finite integer fields, a technique that has shown itself to be effective in simplifying the reduction process and reducing computational overhead. By leveraging this method, Kira can isolate linearly independent systems quickly, optimizing the reduction phase without sacrificing computational resources on redundant equations.
To achieve its goals, Kira employs the tools of computer algebra systems and interfacing methods (such as GiNaC and Fermat) to handle the intricate symbolic manipulations required for Feynman integral reductions. The triangulation and subsequent reduction phases are executed through straightforward Gauss-type elimination processes, adeptly culminating in the expression of all seed integrals in terms of master integrals without generating excessive intermediate complexity.
The paper presents benchmarks conducted with Kira on double box topologies relevant to NNLO corrections for single top-quark production at colliders such as the LHC. These tests demonstrated that Kira exhibits competitive, often superior, performance compared to existing reduction tools like Reduze2 and FIRE5, especially when handling multi-scale problems. For instance, Kira significantly cuts down on computational time in its reduction types by efficiently managing the dimensional algebra involved in the back substitution process.
Implications and Future Directions
The implementation of Kira represents a noteworthy advancement in the field of computational physics, offering a robust tool for the reduction of complicated Feynman diagrams. Its ability to manage intricate dependencies and optimize algebraic simplifications is invaluable for theoretical physicists pushing the boundaries of higher-order loop corrections.
Future developments may focus on expanding Kira's applicability to even higher loop integrals, integrating advanced methods for more efficient symbolic manipulation and further reducing memory consumption. The robust handling of divergent integrals through dimensional regularization and simplifying algebraic multivariate functions underpins the continued evolution towards even more comprehensive and optimized computational frameworks.
In conclusion, Kira stands as a significant contribution to the landscape of Feynman integral reduction tools, characterized by its novel application of modular arithmetic and adept handling of complex equation systems, underlining its potential impact on forthcoming theoretical research and experimentation in particle physics.