Abstract
The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.
Highlights
In recent years there has been significant progress in understanding the mathematical properties of dimensionally-regularised Feynman integrals
It is a diagrammatic coaction, in that it is defined in terms of the topological data of the Feynman graph, and we have argued and provided evidence [1, 2, 20, 50, 51] that it reproduces both the local coaction if each integral is replaced by its Laurent expansion in, and the global coaction if each integral is replaced by the corresponding all-order in expression
The main features of this coaction are as follows: first, if Feynman graphs are replaced by the functions they represent in dimensional regularisation, the diagrammatic coaction maps directly to the global coaction on hypergeometric functions, which in turn agrees with the local coaction acting on multiple polylogarithms (MPLs) order-by-order in the expansion
Summary
In recent years there has been significant progress in understanding the mathematical properties of dimensionally-regularised Feynman integrals. Once a basis of integrands has been chosen, we can construct a corresponding (dual) set of contours and express each element of the global coaction of these master integrals in terms of the same class of hypergeometric functions [20] This set of contours defines a basis for the relevant homology group To this end we consider the example of a one-loop bubble integral with a single massive propagator, which evaluates in dimensional regularisation to a Gauss hypergeometric function. We provide some details on the computation of selected cuts in appendix A, comment on how differential equations can be used to constrain the form of cut integrals in appendix B, and list the expressions for all the master integrals and the corresponding cuts which appear in our coaction examples in appendix C
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