A Feynman integral via higher normal functions
Abstract
We study the Feynman integral for the threebanana graph defined as the scalar twopoint selfenergy at threeloop order. The Feynman integral is evaluated for all identical internal masses in two spacetime dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical PicardFuchs 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 HasseWeil Lfunction of the K3 surface. This result is shown to be a particular case of Deligne's conjectures relating values of Lfunctions inside the critical strip to periods.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.2664
 Bibcode:
 2014arXiv1406.2664B
 Keywords:

 High Energy Physics  Theory;
 High Energy Physics  Phenomenology;
 Mathematical Physics;
 Mathematics  Algebraic Geometry
 EPrint:
 Latex. 70 pages. 3 figures. v3: minor changes and clarifications. Version to appear in Compositio Mathematica