Abstract
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.
Highlights
Feynman integrals are central objects in the perturbative approach to quantum field theory
Since we have argued that all one-loop Feynman integrals, as well as their cuts, can be expressed in terms of multiple polylogarithms (MPLs), we give a short review of the mathematics of MPLs and their algebraic properties
Our main result is the conjecture (4.12), stating that the coaction of eq (4.7), defined purely in terms of graphs, exactly reproduces the combinatorics of the coaction on MPLs when applied to the Laurent coefficients in the expansion of Feynman integrals
Summary
Feynman integrals are central objects in the perturbative approach to quantum field theory. Understanding the mathematics and algebraic structure of MPLs has led both to new efficient techniques to evaluate Feynman integrals [3,4,5,6,7,8,9] and to tools to handle the complicated analytic expressions inherent to these computations [10] It is known, that starting from two loops there are Feynman integrals that cannot be written in terms of MPLs only, and generalizations to elliptic curves appear [11,12,13,14,15,16,17,18,19,20,21,22]. We include several appendices where we summarize the notation and review some mathematical concepts used throughout the paper, and we present explicit results for the (cut) integrals
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