Abstract
The R*-operation by Chetyrkin, Tkachov, and Smirnov is a generalisation of the BPHZ R-operation, which subtracts both ultraviolet and infrared divergences of euclidean Feynman graphs with non-exceptional external momenta. It can be used to compute the divergent parts of such Feynman graphs from products of simpler Feynman graphs of lower loops. In this paper we extend the R*-operation to Feynman graphs with arbitrary numerators, including tensors. We also provide a novel way of defining infrared counterterms which closely resembles the definition of its ultraviolet counterpart. We further express both infrared and ultraviolet counterterms in terms of scaleless vacuum graphs with a logarithmic degree of divergence. By exploiting symmetries, integrand and integral relations, which the counterterms of scaleless vacuum graphs satisfy, we can vastly reduce their number and complexity. A FORM implementation of this method was used to compute the five loop beta function in QCD for a general gauge group. To illustrate the procedure, we compute the poles in the dimensional regulator of all top-level propagator graphs at five loops in four dimensional ϕ3 theory.
Highlights
These divergences are known as collinear and soft divergences and are often collectively called infrared (IR) divergences
That we have defined the appropriate notations, we describe the conditions for an IR subgraph to be IR irreducible (IRI) [29]: (i) No external momentum flows into an internal vertex of γ′, (ii) γ′ cannot contain massive lines, (iii) the associated contracted graph γcannot contain cut-vertices,2 (iv) [for insertions] each connected component in the remaining graph γshould be 1PI after shrinking massive lines and welding the vertices together that have external momenta attached to it
The R∗-operation is a powerful tool to compute the poles of arbitrary euclidean Feynman graphs with non-exceptional external momenta from simpler Feynman graphs
Summary
We impose that all Feynman graphs to be considered in the following are euclidean and will always have non-exceptional external momenta. To be precise, this means that no linear combination of external momenta p1, . Pn vanishes: pi = 0, i∈I for I any subset of {1, . The divergences which exist in non-exceptional Feynman graphs can be classed into two types:. (ii) IR divergences, related to vanishing loop momentum configurations. In the following we shall review the basic notions of power counting for UV and IR divergences and thereby introduce the necessary language which will be needed later to define the R- and R∗-operations
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