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

“…It also confirms the leading and subleading terms in the large N expansion of the 6-loop beta function obtained almost 20 years ago in [19,35]. Additional checks were provided by Dmitrii Batkovich, who calculated 575 out of the 627 six-loop Γ 4 -counterterms using the traditional approach, which combines IBP for four-loop massless propagators (known from [2,43,61]) and IRR with R * to compute the diagrams [5]. The most complicated diagrams (that could not be checked this way) were additionally computed numerically with sector decomposition [9], using a custom-made program by the first author, to at least 3 significant digits.…”

“…5 The operator K determines the renormalization scheme and is given by (3.2) for MS. While (5.1) guarantees that all poles in ε cancel in the sum of the regularized integrals, we do not obtain subdivergence-free integrands this way.…”

“…5 Remarkably, this is essentially just a formula for the antipode in the Hopf algebra of Feynman graphs [42].…”

“…The comparison with Oliver's result convinced us of the correctness, since the only discrepancy was localized in the coefficient of ζ 9 and could be tracked down to a copy error when we transferred the results for the primitive graphs from [57] into our program. 7 Very valuable checks of plenty of 6-loop integrals via the IBP/IRR/R * -method were provided by Dmitrii Batkovich [5].…”

Introducing Loopedia

“…A more detailed description of the Nickel Index can be found in the paper of Batkovich et al [8], where they introduce the GraphState library. It is used by Loopedia to manipulate Nickel Indices.…”