Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals (1403.3385v1)

Abstract: We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we discuss various applications. In particular, many Feynman integrals can be computed by this method.

• The paper introduces a novel algorithm for symbolically integrating hyperlogarithms, enhancing the computation of multi-loop Feynman integrals.
• It employs a three-step process—function decomposition, primitive calculation, and limit evaluation—to handle divergent integrals effectively.
• The implementation incorporates dimensional regularization and polynomial reduction, providing a practical tool for quantum field theory research.

Overview of Symbolic Integration of Hyperlogarithms for Feynman Integrals

The discussed paper presents algorithms for the symbolic integration of hyperlogarithms, focusing on applications to Feynman integrals. This work contributes significantly to the field of computational techniques in theoretical physics, particularly in the evaluation of multi-loop Feynman integrals using the parametric representation. Feynman integrals are essential in quantum field theory as they describe the behavior of particles and fields.

Core Contributions

The core contribution is an algorithm for integrating hyperlogarithms symbolically, which also encompasses multiple polylogarithms when their arguments are rational functions. These algorithms are grounded in the nature of hyperlogarithms as iterated integrals. Such functions naturally arise in the resolution of complex integrals and have applications beyond physics, extending to combinatorics and pure mathematics. The software implementation of these algorithms enables researchers to compute Feynman integrals more effectively, potentially leading to better theoretical predictions in particle physics.

Algorithmic Strategy

The paper details a three-step strategy for integrating functions that are hyperlogarithms in one variable:

1. Function Decomposition: The integrand is first expressed as a hyperlogarithm in terms of the variable to be integrated.
2. Primitive Calculation: A primitive function is found for the integrand using the iterated integral structure inherent in hyperlogarithms.
3. Limit Evaluation: The definite integral is computed as the difference of limits at infinity and zero of the primitive function.

The uniqueness of this approach lies in its treatment of divergent integrals. The authors introduce a method for regularizing such limits, employing transformation properties of hyperlogarithms, thereby extending the applicability to a broader class of Feynman integrals.

Numerical Implications and Examples

The paper includes robust examples illustrating the application of these techniques on genuine physical models, such as multi-loop integrals common in quantum field computations. Notably, the algorithms handle dimensional regularization via a differential expansion concerning the parameter $\varepsilon$, facilitating the evaluation of divergent Feynman integrals. The implementation addresses challenges such as factorization over polynomial remainders, essential for treating non-linear polynomials as effectively linear.

Limitations and Future Directions

A noteworthy restriction of the method is its dependence on the linear reducibility of the integrals. The paper provides a framework utilizing polynomial reduction strategies to ascertain this reducibility, thereby guiding the selection of computational paths. While linear reducibility limits the direct applicability of the method to certain classes of graphs, future research may explore expansions or modifications to handle a wider variety of integrals, including those that exceed the domain of polylogarithmic evaluations.

From a practical perspective, the paper envisions further development in symbolic computation, encouraging the integration of these algorithms into more advanced computational tools used in physics and applied mathematics. The documentation suggests potential interplay with other mathematical software, offering a resource for theoretical and computational studies in quantum field theory where high-precision calculations are paramount.

Conclusion

This work provides a significant methodological framework for the symbolic evaluation of Feynman integrals, reinforcing the vital role of hyperlogarithms. The algorithms serve as a critical toolset for theoretical physicists seeking to solve integrals that model complex particle interactions. As the field advances, the principles demonstrated here may guide future innovations that manage to bridge computational efficiency with theoretical complexity across various areas of mathematical physics.