A Feynman integral via higher normal functions (1406.2664v3)

Abstract: We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of K3 surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the K3 family. We prove a conjecture by David Broadhurst that at a special kinematical point the Feynman integral is given by a critical value of the Hasse-Weil L-function of the K3 surface. This result is shown to be a particular case of Deligne's conjectures relating values of L-functions inside the critical strip to periods.

• The paper demonstrates that the Feynman integral is derived as an inhomogeneous solution to a Picard-Fuchs differential equation, linking algebraic cycles with period integrals.
• It employs motivic cohomology and Eisenstein series to show that at specific kinematical points the three-loop integral matches critical L-function values, including 7ζ(3) at t=0.
• The interdisciplinary approach paves the way for further exploration of quantum field computations and the deep connection between arithmetic geometry and physics.

Overview: A Feynman Integral via Higher Normal Functions

Introduction

The paper "A Feynman Integral via Higher Normal Functions" by Bloch, Kerr, and Vanhove, investigates the evaluation of the Feynman integral associated with the three-banana graph. The integral in question is a scalar two-point self-energy at three-loop order in a two-dimensional space-time setting, computed for all identical internal masses. This paper is notable for employing an interdisciplinary approach, utilizing both mathematical techniques such as Picard-Fuchs differential equations and motivic cohomology, and concepts from physics, such as Eisenstein series.

Methodology and Calculation

The paper provides two primary methods for calculating the Feynman integral:

1. Differential Equation Approach: The authors interpret the integral as an inhomogeneous solution to a classical Picard-Fuchs differential equation. This method capitalizes on the fact that the family of Feynman integrals can be linked to higher algebraic cycles and examined using the variation of Hodge structures. The paper details the computation of an inhomogeneous Picard-Fuchs equation satisfied by the three-banana integral, thus offering evidence for the relationship between these integrals and periods.
2. Arithmetic Algebraic Geometry Approach: Utilizing motivic cohomology, the authors view the Feynman integral as a family of regulator periods associated with a family of $K3$ surfaces. This enriches the understanding through the use of Eisenstein series and led to proving David Broadhurst's conjecture that at a certain kinematical point, the Feynman integral equates to a critical value of the Hasse-Weil $L$-function of the $K3$ surface.

Key Results and Implications

The Feynman integral is expressed as a sum of elliptic trilogarithms evaluated at sixth roots of unity, with the trilogarithm correlated to the regulator of a class within the motivic cohomology of the $K3$ family. This connection elucidates Deligne's conjectures linking $L$-function values within the critical strip to periods.

The analysis unveils that the three-banana integral at $t = 0$ yields a previously obtained numerical result of $7\zeta(3)$. Furthermore, at $t = 1$, the integral represents a rational period of the $K3$ surface connected to the critical value of the Hasse-Weil $L$-function. These special values exemplify the potential for using higher algebraic cycles in predicting critical values of $L$-functions.

Speculations and Future Directions

The implications of these findings are vast, connecting algebraic geometry concepts, such as $K3$ surfaces, with deep results in number theory regarding $L$-functions. This alliance encourages further exploration in both the applied physics field, as well as the theoretical mathematics field, heralding new insights into quantum field theory computations and their mathematical underpinnings.

Given the robust framework established in this paper, future work could explore the potential of similar methods in calculating Feynman integrals for other multi-loop graphs, thereby continuing to bridge the gap between physics and arithmetic algebraic geometry. The paper also suggests directions for further research on motivic normal functions and their applications in physics modeling, such as mirror symmetry and string theory.

The paper’s integration of geometric, arithmetic, and physical methodologies presents a promising addition to the literature, offering rich avenues for expanding both the foundational understanding and applied aspects of Feynman integrals.