Multiloop integrals in dimensional regularization made simple (1304.1806v2)

Abstract: Scattering amplitudes at loop level can be expressed in terms of Feynman integrals. The latter satisfy partial differential equations in the kinematical variables. We argue that a good choice of basis for (multi-)loop integrals can lead to significant simplifications of the differential equations, and propose criteria for finding an optimal basis. This builds on experience obtained in supersymmetric field theories that can be applied successfully to generic quantum field theory integrals. It involves studying leading singularities and explicit integral representations. When the differential equations are cast into canonical form, their solution becomes elementary. The class of functions involved is easily identified, and the solution can be written down to any desired order in epsilon within dimensional regularization. Results obtained in this way are particularly simple and compact. In this letter, we outline the general ideas of the method and apply them to a two-loop example.

• The paper establishes that selecting an optimal basis significantly simplifies the evaluation of complex multiloop integrals.
• It reformulates the differential equations into a canonical form, enabling computation to any order in dimensional regularization.
• The uniform, compact solutions are validated via two-loop examples, enhancing precision in quantum field theory calculations.

Multiloop Integrals in Dimensional Regularization Made Simple

The paper entitled "Multiloop integrals in dimensional regularization made simple" by Johannes M. Henn, explores advancements in simplifying the evaluation of scattering amplitudes at loop levels through the strategic selection of a basis for Feynman integrals. This work provides substantial progress in computing these complex integrals, which play a critical role in the domain of quantum field theory.

Overview

Scattering amplitudes are fundamental in connecting theoretical predictions with experimental observations in quantum field theory. These computations are notoriously complex, especially at loop levels where multiloop integrals become involved. The paper identifies that the complexity in solving these integrals can be significantly reduced by adopting an optimal choice of basis. The research builds upon insights from supersymmetric field theories, suggesting that its results are applicable to more general quantum field theories. The simplification is achieved by writing the differential equations that these integrals satisfy in a "canonical" form.

Key Contributions

1. Identifying an Optimal Basis: The paper sets forth criteria for determining a suitable basis for multiloop integrals, focusing on simplifying the accompanying differential equations. This optimal choice hinges on the concept of leading singularities and the properties of iterated integrals, hinting at a deeper mathematical structure underlying these results.
2. Form of Differential Equations: By casting differential equations into a canonical form, they become more straightforwardly solvable. The class of functions that appear in these solutions is easily identifiable, and these solutions can be computed to any desired order in the parameter $\epsilon$ within dimensional regularization.
3. Compact and Simple Results: The solutions obtained through this method are notably compact and retain uniform transcendentality. The practical value of these solutions is underscored by their application to a two-loop example, demonstrating the efficacy of the approach.

Implications

The implications of this research are twofold. Practically, it enhances the precision and efficiency with which multiloop integrals in quantum field theories are computed. Theoretically, it provides insights into the intrinsic structures of these calculations, potentially unveiling more profound connections in quantum field theories.

The paper also opens avenues for future work. It suggests further exploration into optimal basis choices across various quantum field theories, including those with massive particles like top quarks or Higgs bosons. Additionally, the approach could be applicable to theories beyond four dimensions, suggesting that the framework developed could have broad applicability.

Numerical Results and Future Directions

The numerical results demonstrate the theory’s capabilities through a set of two-loop planar integrals. The authors show that each integral solution displays uniform degree of transcendentality, aligning with the proposed theoretical framework. These integrals, pertinent to massless $2\to2$ scattering processes, exemplify the potential predictive power harnessed through an optimized basis selection.

Moving forward, it would be valuable to explore the identification of suitable integral functions for more complex processes involving strong coupling or massive particles. Moreover, understanding how these techniques can interact with symbolic manipulation tools could further streamline computations in multiloop integral evaluations.

Conclusion

This paper makes a significant stride in the methodology for evaluating multiloop integrals within the context of quantum field theory. Its insights into basis selection and differential equation simplification promise more accessible and precise computations in dimensional regularization, which is crucial for advancing theoretical physics and its experimental correspondences. Future research will undoubtedly extend these concepts across a broader spectrum of applications, fostering a deeper understanding of the complex interactions delineated by theoretical physics.