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We compute the two-loop $\beta$-function of scalar and spinorial quantumelectrodynamics as well as pure Yang-Mills and quantum chromodynamics using thebackground field method in a fully quadridimensional setup using ImplicitRegularization (IREG). Moreover, a thorough comparison with dimensionalapproaches such as conventional dimensional regularization (CDR) anddimensional reduction (DRED) is presented. Subtleties related to Lorentzalgebra contractions/symmetric integrations inside divergent integrals as wellas renormalisation schemes are carefully discussed within IREG where therenormalisation constants are fully defined as basic divergent integrals toarbitrary loop order. Moreover, we confirm the hypothesis that momentum routinginvariance in the loops of Feynman diagrams implemented via settingwell-defined surface terms to zero deliver non-abelian gauge invariantamplitudes within IREG just as it has been proven for abelian theories.

Two-loop renormalisation of gauge theories in $4D$ Implicit Regularisation and connections to dimensional methods