Abstract
We review Zimmermann's forest formula, which solves Bogoliubov's recursive R-operation for the subtraction of ultraviolet divergences in perturbative Quantum Field Theory. We further discuss a generalisation of the R-operation which subtracts besides ultraviolet also Euclidean infrared divergences. This generalisation, which goes under the name of the R⁎-operation, can be used efficiently to compute renormalisation constants. We will discuss several results obtained by this method with focus on the QCD beta function at five loops as well as the application to hadronic Higgs boson decay rates at N4LO. This article summarizes a talk given at the Wolfhart Zimmermann Memorial Symposium.
Highlights
Despite the enormous success in describing the interactions of elementary particles the appearance of ultraviolet (UV) divergences make it difficult to establish Quantum Field Theory as a fundamental theory of nature
A solution which at least grants the interpretation of Quantum Field Theories as low energy effective theories is given by the procedure of renormalisation
The R∗-operation should be regarded more as a mathematical trick - rather than a renormalisation scheme - which allows one to extract the renormalisation constants of Feynman integrals or correlators from maximally simple onescale Feynman Integrals. It achieves this by making use of the technique of IR rearrangement (IRR) [12] in dimensional regularisation
Summary
Despite the enormous success in describing the interactions of elementary particles the appearance of ultraviolet (UV) divergences make it difficult to establish Quantum Field Theory as a fundamental theory of nature. An important development in the establishment of renormalisation theory has been the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalisation scheme This scheme was originally developed by Bogoliubov and Parasiuk [1] in terms of a recursive subtraction operation, often called Bogoliubov’s R-operation. This method of renormalisation makes it possible to subtract the complicated overlapping and nested UV-divergences which can appear in Feynman integrals, the building blocks of the perturbative expansion. An alternative proof for the finiteness of the renormalised Feynman Integral was given by Zimmermann [3, 4] He realised in particular that the recursion in Bogoliubov’s R-operation gives rise to a sum over forests of graphs.
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