Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

Samuel Abreu (1) , Ruth Britto (2, 3, 4) , Claude Duhr (5, 6) , Einan Gardi (7)

1 Albert-Ludwigs-Universität Freiburg
2 Trinity College Dublin
3 IPHT - Institut de Physique Théorique - UMR CNRS 3681
4 KITP - Kavli Institute for Theoretical Physics [Santa Barbara]
5 CERN [Genève]
6 UCL - Université Catholique de Louvain = Catholic University of Louvain
7 Edin. - University of Edinburgh

We construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our conjecture holds in a broad range of nontrivial one-loop integrals. We then explore its consequences for the study of discontinuities of Feynman integrals, and the differential equations that they satisfy. In particular, using the diagrammatic coaction along with information from cuts, we explicitly derive differential equations for any one-loop Feynman integral. We also explain how to construct the symbol of any one-loop Feynman integral recursively. Finally, we show that our diagrammatic coaction follows, in the special case of one-loop integrals, from a more general coaction proposed recently, which is constructed by pairing master integrands with corresponding master contours.

Mots clés: Feynman integrals; cuts; polylogarithms; Hopf algebra

Domaines: Physique [physics]

Samuel Abreu, Ruth Britto, Claude Duhr, Einan Gardi. Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case. Journal of High Energy Physics, 2017, 2017, pp.90. ⟨10.1007/JHEP12(2017)090⟩. ⟨cea-01518887⟩