Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 048, 12 pages      arXiv:2105.06132      https://doi.org/10.3842/SIGMA.2021.048
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Double Box Motive

Spencer Bloch
Department of Mathematics, The University of Chicago, Eckhart Hall, 5734 S University Ave, Chicago IL, 60637, USA

Received March 20, 2021, in final form May 04, 2021; Published online May 13, 2021

Abstract
The motive associated to the second Symanzik polynomial of the double-box two-loop Feynman graph with generic masses and momenta is shown to be an elliptic curve.

Key words: Feynman amplitude; elliptic curve; double-box graph; cubic hypersurface.

pdf (345 kb)   tex (16 kb)  

References

1. Bloch S., Vanhove P., The elliptic dilogarithm for the sunset graph, J. Number Theory 148 (2015), 328-364, arXiv:1309.5865.
2. Collino A., The Abel-Jacobi isomorphism for the cubic fivefold, Pacific J. Math. 122 (1986), 43-55.
3. Deligne P., Équations différentielles à points singuliers réguliers,Lecture Notes in Math., Vol. 163, Springer-Verlag, Berlin - New York, 1970.
4. Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York - Heidelberg, 1977.
5. Huybrechts D., The geometry of cubic hypersurfaces, Preprint, 2020.
6. Kreimer D., The master two-loop two-point function. The general case, Phys. Lett. B 273 (1991), 277-281.
7. Peters C.A.M., Steenbrink J.H.M., Mixed Hodge structures, textitErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 52, Springer-Verlag, Berlin, 2008.