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The five-loop beta function of Yang-Mills theory with fermions

A preprint version of the article is available at arXiv.


We have computed the five-loop corrections to the scale dependence of the renormalized coupling constant for Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories with a simple compact Lie group, and for Quantum Electrodynamics (QED). Our analytical result, obtained using the background field method, infrared rearrangement via a new diagram-by-diagram implementation of the R* operation and the Forcer program for massless four-loop propagators, confirms the QCD and QED results obtained by only one group before. The numerical size of the five-loop corrections is briefly discussed in the standard \( \overline{\mathrm{MS}} \) scheme for QCD with n f flavours and for pure SU(N) Yang-Mills theory. Their effect in QCD is much smaller than the four-loop contributions, even at rather low scales.


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Herzog, F., Ruijl, B., Ueda, T. et al. The five-loop beta function of Yang-Mills theory with fermions. J. High Energ. Phys. 2017, 90 (2017).

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  • Perturbative QCD
  • Renormalization Group