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A Feynman integral via higher normal functions

Spencer Bloch, Matt Kerr, Pierre Vanhove

TL;DR

The paper unifies quantum-field-theory amplitudes with deep arithmetic geometry by analyzing the three-banana Feynman integral in 2D as a regulator period of a K3-family. It develops two complementary computations: a PF-differential approach and a motivic/arithmetic-geometric method using higher Chow groups, Abel-Jacobi theory, and Eisenstein symbols, yielding an explicit expression in terms of K3 periods and elliptic trilogarithms. The authors prove a Broadhurst-type L-value relation at t=1 in line with Deligne's conjectures, and establish exact special values I(0)=7ζ(3) and I(1) up to rational factors, thereby revealing deep ties between Feynman amplitudes, regulator currents, and motives. The results illuminate how motivic normal functions and modular geometry govern amplitudes, suggesting broad connections between quantum field theory and arithmetic geometry.

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.

A Feynman integral via higher normal functions

TL;DR

The paper unifies quantum-field-theory amplitudes with deep arithmetic geometry by analyzing the three-banana Feynman integral in 2D as a regulator period of a K3-family. It develops two complementary computations: a PF-differential approach and a motivic/arithmetic-geometric method using higher Chow groups, Abel-Jacobi theory, and Eisenstein symbols, yielding an explicit expression in terms of K3 periods and elliptic trilogarithms. The authors prove a Broadhurst-type L-value relation at t=1 in line with Deligne's conjectures, and establish exact special values I(0)=7ζ(3) and I(1) up to rational factors, thereby revealing deep ties between Feynman amplitudes, regulator currents, and motives. The results illuminate how motivic normal functions and modular geometry govern amplitudes, suggesting broad connections between quantum field theory and arithmetic geometry.

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.

Paper Structure

This paper contains 29 sections, 20 theorems, 282 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The three-banana Feynman integral
  3. The integral
  4. The Picard-Fuchs equation
  5. Solution of the inhomogeneous Picard-Fuchs equation
  6. Value of the integral at $t=0$
  7. Value of the integral at $t=1$
  8. The family of $K3$ surfaces
  9. Sunset in a nutshell
  10. Verrill's family
  11. Miscellany
  12. The three-banana integral as a higher normal function
  13. $\mathbf{K_{3}}$ of a $\mathbf{K3}$!
  14. Review of Abel-Jacobi
  15. Reinterpreting the Feynman integral
  16. ...and 14 more sections

Key Result

Theorem 2.1.1

The integral $I_{\textlinb{\Bqe}}(m_i;t)$ defined in eq e:I3masses has the following integral representation for $t<(\sum_{i=1}^4 m_i)^2$

Figures (1)

  • Figure 2.1.1: The three-loop three-banana Feynman graph. $K$ is the external momentum in ${\mathbb R}^{2}$ and $m_i\geq0$ with $i=1,\dots,4$ are internal masses.

Theorems & Definitions (47)

  • Theorem 2.1.1
  • proof
  • Theorem 2.2.1
  • proof
  • Lemma 2.2.2
  • proof
  • Lemma 2.3.1
  • proof
  • Theorem 2.3.2
  • proof
  • ...and 37 more