Feynman integrals and hyperlogarithms

Abstract: We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional) regularization are summarized and we prove that in the Euclidean region, each Feynman integral can be written as a linear combination of convergent Feynman integrals. This means that one can choose a basis of convergent master integrals and need not evaluate any divergent Feynman graph directly. Secondly we give a self-contained account of hyperlogarithms and explain in detail the algorithms needed for their application to the evaluation of multivariate integrals. We define a new method to track singularities of such integrals and present a computer program that implements the integration method. As our main result, we prove the existence of infinite families of massless 3- and 4-point graphs (including the ladder box graphs with arbitrary loop number and their minors) whose Feynman integrals can be expressed in terms of multiple polylogarithms, to all orders in the epsilon-expansion. These integrals can be computed effectively with the presented program. We include interesting examples of explicit results for Feynman integrals with up to 6 loops. In particular we present the first exactly computed counterterm in massless phi^4 theory which is not a multiple zeta value, but a linear combination of multiple polylogarithms at primitive sixth roots of unity (and divided by $\sqrt{3}$). To this end we derive a parity result on the reducibility of the real- and imaginary parts of such numbers into products and terms of lower depth.

An Analysis of Feynman Integrals and Hyperlogarithms by Erik Panzer

Erik Panzer's comprehensive work on "Feynman Integrals and Hyperlogarithms" constitutes a significant mathematical investigation into the complex nature of Feynman integrals, a key component of perturbative quantum field theory. This publication, which builds upon his PhD thesis, thoroughly explores the mathematical structures underpinning these integrals and presents algorithms and methodologies for their evaluation.

Overview of Feynman Integrals and Polylogarithms

Feynman integrals are an integral part of calculating scattering amplitudes, which are essential in the prediction of particle interactions. Panzer's work begins by addressing the challenge posed by these integrals due to their intricate dependencies on various parameters such as particle masses and external momenta. He leverages techniques from analytic regularization, Schwinger parameters, and the hyperlogarithm framework to derive a structured approach to handling these complexities. The paper establishes that, within the Euclidean region, each Feynman integral can be expressed as a linear combination of convergent master integrals, thus circumventing the need to directly compute any divergent graphs.

Hyperlogarithms: Algorithmic Integration and Computation

Central to Panzer's thesis is the employment of hyperlogarithms in the computation of these integrals. This mathematical tool is adept at addressing the multifaceted nature of Feynman integrals, which frequently involve iterated integrals over rational functions. Panzer introduces an effective method to track and manage singularities, thereby enhancing the utility of hyperlogarithms in evaluating multivariate integrals. The HyperInt program, also developed as part of this research, is a critical outcome, providing an automation tool for these complex evaluations.

Recursive Calculations and Explicit Results

Panzer offers explicit examples of recursive integral formulas suited to massless 3- and 4-point functions, including sophisticated ladder box graphs of any loop number. Notably, he presents the first precisely calculated counterterm in massless φ-theory that deviates from being just a multiple zeta value, highlighting additional layers of complexity in Feynman integrals. This discovery involves multiple polylogarithms at primitive sixth roots of unity, showcasing the novel results derived from Panzer's methods.

Implications and Future Directions

Panzer's work not only advances the theoretical comprehension of Feynman integrals but also provides practical tools and perspectives for the computation of complex scattering amplitudes. The implications for both perturbative quantum field theory and computational algebra are significant, promising improved methodologies for handling divergences in theoretical physics computations.

In a broader context, Panzer's exploration opens new pathways in the study of hypergeometric functions and their applications in physics, suggesting that further development of polynomial reduction techniques and the understanding of linear reducibility could yield even more efficient computational methods. This extends the reach of hyperlogarithmic computations beyond Feynman integrals, potentially impacting other domains in theoretical physics and mathematics.

Conclusion

Erik Panzer's paper is a profound contribution to the field of mathematical physics, offering both theoretical insights and practical solutions to the challenge of evaluating Feynman integrals. His innovative use of hyperlogarithms, coupled with the development of computational tools and recursive methods, underscores a substantial step forward in the analytic techniques available for perturbative quantum field theory. This work not only addresses existing challenges but also sets the stage for future research and development in related areas of mathematical exploration.