Toolbox for multiloop Feynman diagrams calculations usingR* operation

Abstract

We present the toolbox for analytical calculation of U V -counterterm of Feynman diagrams. It combines the power of R * -operation with modern analytical methods. Written in pure Python our toolbox can be easily used and extended.

Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops

“…The main program is written in Python using the GraphState/Graphine library, which provide a very convenient way to manipulate Feynman graphs [4,6]. It generates the graphs, computes their symmetry factors and combines them into the counterterms (3.1) to calculate the anomalous dimensions and the beta function via (2.5) and (2.4).…”

“…Since U is linear in all α e , the first integration is elementary and the result can be interpreted as the p-integral G \ e (with the external momenta entering at the vertices that were incident to e) [21,48]. 6 All 4-point 6-loop graphs contributing to Γ 4 are therefore effectively expressed in terms of 5-loop p-integrals. In the traditional approach, this simplification is achieved with the R * -operation [24,26,30,32].…”

“…The general philosophy has hitherto been to exploit various relations, in particular the infrared R * -operation [24,26,30,32], to relate the counterterms to simpler integrals. Recently, the cumbersome R * -operation was automated in [6]. Augmented with the results [2,43,61] for 4-loop p-integrals and integration by parts (IBP) [31,65], this traditional approach was used for the 6-loop calculation of the field anomalous dimension [7].…”

On Higgs decays to hadrons and the R-ratio at N4LO

“…Our use of this approach has been made possible by the development of (a) Forcer [26][27][28], a Form [29][30][31] program for the parametric reduction of four-loop self-energy integrals, and (b) a program [32] efficiently implementing the R * operation, see refs. [33][34][35][36][37], locally for the evaluation of L -loop pole terms in terms of (L −1)-loop integrals. In order to cope with the computations for H → gg, which are far more demanding than those required to determine the five-loop beta function [38,39], the latter program has undergone substantial modifications and extensions.…”

Field theoretic renormalization study of reduced quantum electrodynamics and applications to the ultrarelativistic limit of Dirac liquids

“…111 The power of the BPHZ prescription, with respect to conventional renormalization, lies in the fact that it applies diagram by diagram, gives a very clear and unambiguous prescription on how to subtract (sub-)divergences and is conveniently automated. 84,85 We will apply such a prescription to computing the interaction correction to the optical conductivity and the anomalous dimension of the fermion field (which plays an important role with respect to the critical coupling constant for dynamical gap generation). As will be shown in detail in the following, the obtained results are in complete agreement with those obtained via conventional renormalization thereby lifting any possible ambiguity as to their value at the IR fixed point.…”