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

We construct a diagrammatic coaction acting on one-loop Feynman graphsand their cuts. The graphs are naturally identified with the corresponding (cut) Feynmanintegrals in dimensional regularization, whose coefficients of the Laurent expansion in thedimensional regulator are multiple polylogarithms (MPLs). Our main result is the con-jecture that this diagrammatic coaction reproduces the combinatorics of the coaction onMPLs order by order in the Laurent expansion. We show that our conjecture holds in abroad range of nontrivial one-loop integrals. We then explore its consequences for the studyof discontinuities of Feynman integrals, and the differential equations that they satisfy. Inparticular, using the diagrammatic coaction along with information from cuts, we explicitlyderive differential equations for any one-loop Feynman integral. We also explain how toconstruct the symbol of any one-loop Feynman integral recursively. Finally, we show thatour diagrammatic coaction follows, in the special case of one-loop integrals, from a moregeneral coaction proposed recently, which is constructed by pairing master integrands withcorresponding master contours